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ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2009, Vol. 30, No. 4, pp. 292–326. c© Pleiades Publishing, Ltd., 2009.

Integrable Systems and Complex Geometry

A. Lesfari*

(Submitted by V.V. Lychagin)

Department of Mathematics Faculty of Sciences University of Chouaib Doukkali B.P. 20,El-Jadida, MoroccoReceived July 18, 2009

Abstract—In this paper, we discuss an interaction between complex geometry and integrablesystems. Section 1 reviews the classical results on integrable systems. New examples of integrablesystems, which have been discovered, are based on the Lax representation of the equations of motion.These systems can be realized as straight line motions on a Jacobi variety of a so-called spectralcurve. In Section 2, we study a Lie algebra theoretical method leading to integrable systems andwe apply the method to several problems. In Section 3, we discuss the concept of the algebraiccomplete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the systemis completely integrable in the sens of the phase space being folited by tori, which in addition arereal parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how todecide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in Section 4we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifoldinvariant by the complex flow is covering of abelian variety.

2000 Mathematics Subject Classification: 37J35, 70H06, 14H40, 14H70, 14M10.

DOI: 10.1134/S1995080209040088

Key words and phrases: Completely integrable systems, topological structure of phase space,integration methods, Jacobians, Prym varieties, Complete intersection.

CONTENTS

1. Integrable systems 293

2. Isospectral deformation method 297

2.1. The Euler rigid body motion 299

2.2. The geodesic flow for a left invariant metric on SO(4) 302

2.3. The Toda lattice 305

2.4. The Garnier potential 306

2.5. The coupled nonlinear Schrodinger equations 307

2.6. The Yang–Mills equations 308

*E-mail: [email protected]

292

INTEGRABLE SYSTEMS AND COMPLEX GEOMETRY 293

3. Algebraic complete integrability 308

3.1. A five-dimensional system 312

3.2. The Henon-Heiles system 317

3.3. The Kowalewski rigid body motion 320

3.4. Kirchhoff’s equations of motion of a solid in an ideal fluid 321

4. Generalized algebraic completely integrable systems 323

1. INTEGRABLE SYSTEMS

Let M be an even-dimensional differentiable manifold. A symplectic structure (or symplectic form)on M is a closed non-degenerate differential 2-form ω defined everywhere on M . The non-degeneracycondition means that

∀x ∈ M,∀ξ �= 0,∃η : ω (ξ, η) �= 0, (ξ, η ∈ TxM) .

The pair (M,ω) is called a symplectic manifold.

Example 1.1. The cotangent bundle T ∗M possesses in a natural way a symplectic structure. Ina local coordinate (x1, . . . , xn, y1, . . . , yn), 2n = dim M , the form ω is given by ω =

∑nk=1 dxk ∧ dyk.

Example 1.2. Another important class of symplectic manifolds consists of the coadjointsorbits O ⊂ G∗, where G is the algebra of a Lie group G and Gμ = {Ad∗gμ : g ∈ G} is the orbit ofμ ∈ G∗ under the coadjoint representation.

Theorem 1. (a) Let I: T ∗xM −→ TxM , ω1

ξ −→ ξ, be a map defined by ω1ξ (η) = ω(η, ξ), ∀η ∈ TxM .

Then I is an isomorphism generated by the symplectic form ω.

(b) The symplectic form ω induces a hamiltonian vector field IdH: M −→ TxM , x −→ IdH (x),where H: M −→ R, is a differentiable function (called hamiltonian). In others words, thedifferential system defined by

x(t) = XH (x (t)) = IdH (x) ,

is a hamiltonian vector field associated to the function H . The matrix that is associated to anhamiltonian system determine a symplectic structure.

Proof. (a) Denote by I−1 the map I−1: TxM −→ T ∗xM , ξ −→ I−1 (ξ) ≡ ω1

ξ , with I−1 (ξ) (η) =ω1

ξ (η) = ω (η, ξ), ∀η ∈ TxM . The fact that the form ω is bilinear implies that

I−1 (ξ1 + ξ2) (η) = ω (η, ξ1 + ξ2) ,

= ω (η, ξ1) + ω (η, ξ2) ,

= I−1 (ξ1) (η) + I−1 (ξ2) (η) , ∀η ∈ TxM.

Now, since dim TxM = dim T ∗xM , to show that I−1 is bijective, it suffices to show that is injective. The

form ω is non-degenerate, it follows that

KerI−1 = {ξ ∈ TxM : ω (η, ξ) = 0, ∀η ∈ TxM} = {0} .

Hence I−1 is an isomorphism and consequently I is also an isomorphism (the inverse of an isomorphismis an isomorphism).

(b) Let (x1, . . . , xm) be a local coordinate system on M , (m = dimM). We have

x(t) =n∑

k=1

∂H

∂xkI (dxk) =

n∑

k=1

∂H

∂xkξk, (1)

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 30 No. 4 2009

294 LESFARI

where I (dxk) = ξk ∈ TxM is defined such that: ∀η ∈ TxM , ηk = dxk (η) = ω(η, ξk

), (kth component

of η). Define (η1, . . . , ηm) and(ξk1 , . . . , ξk

m

)to be respectively the components of η and ξk, then

ηk = ω

⎛

⎝m∑

i=1

ηi∂

∂xi,

m∑

j=1

ξkj

∂

∂xj

⎞

⎠

=m∑

i=1

ηi

(∂

∂xi

∂

∂xj

)

ξkj ,

= (η1, . . . , ηm) J−1

⎛

⎜⎜⎜⎜⎝

ξk1

...

ξkm

⎞

⎟⎟⎟⎟⎠

,

where J−1 is the matrix defined by J−1 ≡(

ω

(∂

∂xi,

∂

∂xj

))

1≤i,j≤m

. Since this matrix is invertible,1 we

can search ξk such that:

J−1

⎛

⎜⎜⎜⎜⎝

ξk1

...

ξkm

⎞

⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...

0

1 � k-th place

0...

0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The matrix J−1 is invertible, which implies

⎛

⎜⎜⎜⎝

ξk1...

ξkm

⎞

⎟⎟⎟⎠

= J

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...

0

1

0...

0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

from which ξk = (k-th column of J), i.e., ξki = Jik, 1 ≤ i ≤ m, and consequently ξk =

∑mi=1 Jik

∂

∂xi.

1 Indeed, it suffices to show that the matrix J−1 has maximal rank. Suppose this were not possible, i.e., we assume that

rank(J−1) �= m. Hence∑m

i=1 aiω(

∂∂xi

, ∂∂xj

)= 0, ∀1 ≤ j ≤ m, with ai not all null and ω

(∑mi=1 ai

∂∂xi

, ∂∂xj

)= 0,

∀1 ≤ j ≤ m. In fact, since ω is non-degenerate, we have∑m

i=1 ai∂

∂xi= 0. Now

(∂

∂x1, . . . , ∂

∂xm

)is a basis of TxM ,

then ai = 0, ∀i, contradiction.



It is easily verified that the matrix J is skew-symmetric.2 From (1) we deduce that

x(t) =m∑

k=1

∂H

∂xk

m∑

i=1

Jik∂

∂xi=

m∑

i=1

(m∑

k=1

Jik∂H

∂xk

)∂

∂xi.

Writing x(t) =∑m

i=1

dxi (t)dt

∂

∂xi, it is seen that xi (t) =

∑mk=1 Jik

∂H

∂xk, 1 ≤ i ≤ j ≤ m, which can be

written in more compact form x(t) = J (x)∂H

∂x, this is the hamiltonian vector field associated to the

function H . This concludes the proof of the theorem.We define a Poisson bracket (or Poisson structure) on the space C∞ as

{, } : C∞ (M) × C∞ (M) −→ C∞ (M) , (F,G) −→ {F,G} ,

where {F,G} = duF (XG) = XGF (u) = ω (XG,XF ). This bracket is skew-symmetric {F,G} =−{G,F}, obeys the Leibniz rule {FG,H} = F {G,H} + G {F,H}, and satisfies the Jacobi identity

{{H,F} , G} + {{F,G} ,H} + {{G,H} , F} = 0.

When this Poisson structure is non-degenerate, we obtain the symplectic structure discussed above.Consider now M = R

n ×Rn and let p ∈ M . By Darboux’s theorem [3], there exists a local coordinate

system(x1, . . . , xn, y1, . . . , yn) in a neighbourhood of p such that

{H,F} =n∑

i=1

(∂H

∂xi

∂F

∂yi− ∂H

∂yi

∂F

∂xi

)

.

Then XH =∑n

i=1

(∂H

∂xi

∂

∂yi− ∂H

∂yi

∂

∂xi

)

, and XHF = {H,F}, ∀F ∈ C∞ (M). A nonconstant function

F is called an integral (first integral or constant of motion) of XF , if XHF = 0. In particular, His integral. Two functions F and G are said to be in involution or to commute, if {F,G} = 0. Thehamiltonian systems form a Lie algebra.

We now give the following definition of the Poisson bracket:

{F,G} =⟨

∂F

∂x, J

∂G

∂x

⟩

=∑

i,j

Jij∂F

∂xi

∂G

∂xj.

After some algebraic manipulation, we deduce that If2n∑

k=1

(

Jkj∂Jli

∂xk+ Jki

∂Jjl

∂xk+ Jkl

∂Jij

∂xk

)

= 0, ∀1 ≤ i, j, l ≤ 2n,

then J satisfies the Jacobi identity.Consequently, we have a complete characterization of hamiltonian vector field

x(t) = XH (x (t)) = J∂H

∂x, x ∈ M, (2)

where H : M −→ R, is a differentiable function (the hamiltonian) and J = J (x) is a skew-symmetricmatrix, possibly depending on x ∈ M , for which the corresponding Poisson bracket satisfies the Jacobiidentity:

{{H,F} , G} + {{F,G} ,H} + {{G,H} , F} = 0,

with {H,F} =⟨

∂H

∂x, J

∂F

∂x

⟩

=∑

i,j Jij∂H

∂xi

∂F

∂xj, the Poisson bracket.

2 Indeed, since ω is symmetric i.e., ω

(∂

∂xi, ∂

∂xj

)

= −ω

(∂

∂xj,

∂

∂xi

)

, it follows that J−1 is skew-symmetric. Then,

I = J.J−1 =(J−1

)�.J� = −J−1.J , and consequently J� = J .


296 LESFARI

Example 1.3. An important special case is when J =

⎛

⎜⎝

O −I

I O

⎞

⎟⎠, where I is the n × n identity

matrix. The condition on J is trivially satisfied. Indeed, here the matrix J do not depend on thevariable x and we have

{H,F} =2n∑

i=1

∂H

∂xi

2n∑

j=1

Jij∂F

∂xj=

n∑

i=1

(∂H

∂xn+i

∂F

∂xi− ∂H

∂xi

∂F

∂xn+i

)

.

Moreover, equations (10) are transformed into

q1 =∂H

∂p1, . . . , qn =

∂H

∂pn, p1 = −∂H

∂q1, . . . , pn = −∂H

∂qn,

or q1 = x1, . . . , qn = xn, p1 = xn+1, . . . , pn = x2n. These are exactly the well known differentialequations of classical mechanics in canonical form.

It is a fundamental and important problem to investigate the integrability of hamiltonian systems.Recently there has been much effort given for finding integrable hamiltonian systems, not only becausethey have been on the subject of powerful and beautiful theories of mathematics, but also because theconcepts of integrability have been applied to an increasing number of applied sciences. The so-calledArnold-Liouville theorem play a crucial role in the study of such systems; the regular compact levelmanifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus onwhich the motion is quasi-periodic as a consequence of the following purely differential geometric fact:a compact and connected n-dimensional manifold on which there exist n vector fields which commuteand are independent at every point is diffeomorphic to an n-dimensional real torus and each vector fieldwill define a linear flow there.

Theorem 2. (Arnold-Liouville theorem) [3, 15]: Let H1 = H,H2, . . . ,Hn, be n first integrals ona 2n-dimensional symplectic manifold that are functionally independent (i.e., dH1 ∧ · · · ∧ dHn �=0), and pairwise in involution. For generic c = (c1, . . . , cn) the level set

Mc =n⋂

i=1

{x ∈ M : Hi (x) = ci, ci ∈ R} ,

will be an n-manifold. If Mc is compact and connected, it is diffeomorphic to an n-dimensionaltorus T

n = Rn/Z

n and the solutions of the system (2) are then straight-line motions on Tn. If

Mc is not compact but the flow of each of the vector fields XHkis complete on Mc, then Mc is

diffeomorphic to a cylinder Rk × T

n−k under which the vector fields XHkare mapped to linear

vector fields.

As a consequence, we obtain the concept of complete integrability of a hamiltonian system. For thesake of clarity, we shall distinguish two cases:

(a) Case 1: det J �= 0. The rank of the matrix J is even, m = 2n. A hamiltonian system (2) iscompletely integrable or Liouville-integrable if there exist n firsts integrals H1 = H,H2, . . . ,Hn ininvolution, i.e., {Hk,Hl} = 0, 1 ≤ k, l ≤ n, with linearly independent gradients, i.e., dH1 ∧ · · · ∧ dHn �=0. For generic c = (c1, . . . , cn) the level set

Mc =n⋂

i=1

{x ∈ M : Hi (x) = ci, ci ∈ R} ,

will be an n-manifold. By the Arnold-Liouville theorem, if Mc is compact and connected, it is diffeomor-phic to an n-dimensional torus T

n = Rn/Z

n and each vector field will define a linear flow there. In someopen neighbourhood of the torus there are coordinates s1, . . . , sn, ϕ1, . . . , ϕn in which ω takes the formω =

∑nk=1 dsk ∧ dϕk. Here the functions sk (called action-variables) give coordinates in the direction

transverse to the torus and can be expressed functionally in terms of the firsts integrals Hk. The functionsϕk (called angle-variables) give standard angular coordinates on the torus, and every vector field XHk



can be written in the form ϕk = hk (s1, . . . , sn), that is, its integral trajectories define a conditionally-periodic motion on the torus. In a neighbourhood of the torus the hamiltonian vector field XHk

take thefollowing form sk = 0, ϕk = hk (s1, . . . , sn), and can be solved by quadratures.

(b) Case 2: det J = 0. We reduce the problem to m = 2n + k and we look for k Casimir functions

(or trivial invariants) Hn+1, . . . ,Hn+k, leading to identically zero hamiltonian vector fields J∂Hn+i

∂x= 0,

1 ≤ i ≤ k. In other words, the system is hamiltonian on a generic symplectic manifoldn+k⋂

i=n+1

{x ∈ Rm : Hi (x) = ci} ,

of dimension m − k = 2n. If for most values of ci ∈ R, the invariant manifoldsn+k⋂

i=1

{x ∈ Rm : Hi (x) = ci} ,

are compact and connected, then they are n-dimensional tori Tn = R

n/Zn by the Arnold-Liouville

theorem and the hamiltonian flow is linear in angular coordinates of the torus.

2. ISOSPECTRAL DEFORMATION METHOD

A Lax equation is given by a differential equation of the form

A (t) = [A (t) , B (t)] or [B (t) , A (t)] , (3)

where

A (t) =N∑

k=1

Ak (t) hk, B (t) =N∑

k=1

Bk (t)hk,

are functions depending on a parameter h (spectral parameter) whose coefficients Ak and Bk arematrices in Lie algebras. The pair (A,B) is called Lax pair. This equation established a link betweenthe Lie group theoretical and the algebraic geometric approaches to complete integrability. The solutionto (3) has the form A(t) = g(t)A(0)g(t)−1 , where g(t) is a matrix defined as g (t) = −A(t)g(t). We formthe polynomial P (h, z) = det (A − zI), where z is another variable and I the n × n identity matrix. Wedefine the curve (spectral curve) C, to be the normalization of the complete algebraic curve whose affineequation is P (h, z) = 0.

Theorem 3. The polynomial P (h, z) is independent of t. Moreover, the functions tr (An) arefirst integrals for (3).

Proof. Let us call L ≡ A − zI. Observe that

P = det L.tr(L−1L

)= det L.tr

(L−1BL − B

)= 0,

since trL−1BL = trB. On the other hand

An = AAn−1 + AAAn−2 + · · · + An−1A

= [A,B] An−1 + A [A,B]An−2 + · · · + An−1 [A,B]

= (AB − BA) An−1 + · · · + An−1 (AB − BA)

= ABAn−1 − BAn + · · · + AnB − An−1BA

= A(BAn−1

)−(BAn−1

)A + · · · + A

(An−1B

)−(An−1B

)A.

Since tr (X + Y ) = trX + trY , trXY = trY X, X,Y ∈ Mn (C), we obtain

d

dttr (An

h) = trd

dt(An

h) = 0,


298 LESFARI

and consequently tr (An) are first integrals of motion. This ends the proof of the theorem.

We have shown that a hamiltonian flow of the type (3) preserves the spectrum of A and thereforeits characteristic polynomial. The curve C: P (z, h) = det (A(h) − zI) = 0, is time independent, i.e., itscoefficients tr (An) are integrals of the motion (equivalently, A(t) undergoes an isospectral deformation.Some hamiltonian flows on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Liealgebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs (3). A general statementleading to such situations is given by the Adler-Kostant-Symes theorem.

Theorem 4. Let L be a Lie algebra paired with itself via a nondegenerate, ad-invariant bilinearform 〈 , 〉, L having a vector space decompositionL = K+N with K andN Lie subalgebras. Then,with respect to 〈 , 〉, we have the splitting L = L∗ = K⊥ + N⊥ and N ∗ = K⊥ paired with N via aninduced form 〈〈, 〉〉 inherits the coadjoint symplectic structure of Kostant and Kirillov; its Poissonbracket between functions H1 and H2 on N ∗ reads

{H1,H2} (a) = 〈〈a, [∇N ∗H1,∇N ∗H2]〉〉 , a ∈ N ∗.

Let V ⊂ N ∗ be an invariant manifold under the above co-adjoint action of N on N ∗ and let A(V )be the algebra of functions defined on a neighborhood of V , invariant under the coadjoint actionof L (which is distinct from the N −N ∗ action). Then the functions H in A(V ) lead to commutingHamiltonian vector fields of the Lax isospectral form

a = [a, prK(∇H)] , prK projection onto K.

This theorem produces hamiltonian systems having many commuting integrals ; some preciseresults are known for interesting classes of orbits in both the case of finite and infinite dimensionalLie algebras. Any finite dimensional Lie algebra L with bracket [, ] and killing form 〈, 〉 leads to aninfinite dimensional formal Laurent series extension L =

∑N−∞ Aih

i : Ai ∈ L, N ∈ Z free, with bracket[∑

Aihi,∑

Bjhj]

=∑

i,j [Ai, Bj ]hi+j , and ad-invariant, symmetric forms⟨∑

Aihi,∑

Bjhj⟩k

=∑

i+j=−k 〈Ai, Bj〉, depending on k ∈ Z. The forms 〈, 〉k are non degenerate if 〈, 〉 is so. Let Lp,q (p ≤ q)be the vector space of powers of h between p and q. A first interesting class of problems is obtainedby taking L = Gl(n, R) and by putting the form 〈, 〉1 on the Kac-Moody extension. Then we havethe decomposition into Lie subalgebras L = L0,∞ + L−∞,−1 = K + N with K = K⊥, N = N⊥ andK = N ∗. Consider the invariant manifold Vm, m ≥ 1 in K = N ∗, defined as

Vm =

{

A =m−1∑

i=1

Aihi + αhm, α = diag(α1, · · · , αn) fixed

}

,

with diag (Am−1) = 0.

Theorem 5. The manifold Vm has a natural symplectic structure, the functions H =⟨f(Ah−j), hk

⟩1

on Vm for good functions f lead to complete integrable commuting hamiltoniansystems of the form

A =[A, prK(f ′(Ah−j)hk−j)

], A =

m−1∑

i=0

Aihi + αh,

and their trajectories are straight line motions on the jacobian of the curve C of genus(n − 1)(nm − 2)/2 defined by P (z, h) = det(A − zI) = 0. The coefficients of this polynomialprovide the orbit invariants of Vm and an independent set of integrals of the motion (ofparticular interest are the flows where j = m,k = m + 1 which have the following form

A =[A, adβad−1

α Am−1 + βh], βi = f ′ (αi) ,

the flow depends on f through the relation βi = f ′ (αi) only).

Another class is obtained by choosing any semi-simple Lie algebra L. Then the Kac-Moody exten-sion L equipped with the form 〈, 〉 = 〈, 〉0 has the natural level decomposition L =

∑i∈Z

Li, [Li,Lj ] ⊂



Li+j , [L0, L0] = 0, L∗i = L−i. Let B+ =

∑i≥0 Li and B− =

∑i〈0 Li. Then the product Lie algebra

L × L has the following bracket and pairing[(l1, l2) , (l

′1, l

′2)]

=([l1, l

′1],−[l2, l

′2])

,⟨(l1, l2) , (l

′1, l

′2)⟩

= 〈l1, l′1〉 − 〈l2, l

′2〉.

It admits the decomposition into K + N with

K = {(l,−l) : l ∈ L} , K⊥ = {(l, l) : l ∈ L} ,

N ={(l−, l+) : l− ∈ B−, l+ ∈ B+, pr0(l−) = pr0(l+)

},

N⊥ ={(l−, l+) : l− ∈ B−, l+ ∈ B+, pr0(l+ + l−) = 0

},

where pr0 denotes projection onto L0. Then from the last theorem, the orbits in N ∗=K⊥ possesses a lotof commuting hamiltonian vector fields of Lax form:

Theorem 6. The N-invariant manifolds V−j,k =∑

−j≤i≤k Li ⊆ L � K⊥, has a natural symplec-tic structure and the functions H(l1, l2) = f(l1) on V−j,k lead to commuting vector fields of the Laxform

l =[

l,

(

pr+ − 12pr0

)

∇H

]

, pr+ projection onto B+,

their trajectories are straight line motions on the Jacobian of a curve defined by the characteristicpolynomial of elements in V−j,k.

Using the van Moerbeke-Mumford linearization method [21], Adler and van Moerbeke [1] showedthat the linearized flow could be realized on the jacobian variety Jac(C) (or some sub-abelian variety ofit) of the algebraic curve (spectral curve) C associated to (3). We then construct an algebraic map fromthe complex invariant manifolds of these hamiltonian systems to the jacobian variety Jac(C) of the curveC. Therefore all the complex flows generated by the constants of the motion are straight line motions onthese jacobian varieties i.e. the linearizing equations are given by

s1(t)∫

s1(0)

ωk +

s2(t)∫

s2(0)

ωk + . . . +

sg(t)∫

sg(0)

ωk = ckt, 0 ≤ k ≤ g,

where ω1, . . . , ωg span the g-dimensional space of holomorphic differentials on the curve C of genus g.In an unifying approach, Griffiths [8] has found necessary and sufficient conditions on B for the Laxflow (3) to be linearizable on the jacobi variety of its spectral curve, without reference to Kac-Moody Liealgebras.

Next I schall discuss a number of integrable hamiltonian systems.

2.1. The Euler Rigid Body Motion

It express the free motion of a rigid body around a fixed point. Let M = (m1,m2,m3) be the angularmomentum, Ω = (m1/I1,m2/I2,m3/I3) the angular velocity and I1, I2 et I3, the principal moments ofinertia about the principal axes of inertia. Then the motion of the body is governed by

M = M ∧ Ω. (4)

If one identifies vectors in R3 with skew-symmetric matrices by the rule

a = (a1, a2, a3) , A =

⎛

⎜⎜⎜⎝

0 −a3 a2

a3 0 −a1

−a2 a1 0

⎞

⎟⎟⎟⎠

,


300 LESFARI

then a ∧ b −→ [A,B] = AB − BA. Using this isomorphism between (R3,∧) and (so(3), [, ]), we write(4) as M = [M,Ω], where

M =

⎛

⎜⎜⎜⎝

0 −m3 m2

m3 0 −m1

−m2 m1 0

⎞

⎟⎟⎟⎠

∈ so (3) , Ω =

⎛

⎜⎜⎜⎝

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

⎞

⎟⎟⎟⎠

∈ so (3) .

Now M = IΩ, this implies that

M = [M,ΛM ] , (5)

where

ΛM =

⎛

⎜⎜⎜⎝

0 −λ3m3 λ2m2

λ3m3 0 −λ1m1

−λ2m2 λ1m1 0

⎞

⎟⎟⎟⎠

∈ so (3) ,

with λi ≡ I−1i . Equation (5) is explicitly given by

m1 = (λ3 − λ2)m2m3, m2 = (λ1 − λ3)m1m3, m3 = (λ2 − λ1) m1m2, (6)

and can be written as a hamiltonian vector field

x = J∂H

∂x, x = (m1,m2,m3)

ᵀ ,

with the hamiltonian H =12(λ1m

21 + λ2m

22 + λ3m

23

), and

J =

⎛

⎜⎜⎜⎝

0 −m3 m2

m3 0 −m1

−m2 m1 0

⎞

⎟⎟⎟⎠

∈ so (3) .

We have det J = 0, so m = 2n + k and m − k = rk J . Here m = 3 and rk J = 2, then n = k = 1. The

system (6) has beside the energy H1 = H , a trivial invariant H2, i.e., such that: J∂H2

∂x= 0, or

⎛

⎜⎜⎜⎝

0 −m3 m2

m3 0 −m1

−m2 m1 0

⎞

⎟⎟⎟⎠

⎛

⎜⎜⎜⎝

∂H2∂m1

∂H2∂m2

∂H2∂m3

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎝

0

0

0

⎞

⎟⎟⎟⎠

,

implying∂H2

∂m1= m1,

∂H2

∂m2= m2,

∂H2

∂m3= m3, and consequently

H2 =12(m2

1 + m22 + m2

3

).

The system evolves on the intersection of the sphere H1 = c1 and the ellipsoid H2 = c2. In R3, this

intersection will be isomorphic to two circles(

withc2

λ3< c1 <

c2

λ1

)

. We shall show that the problem

can be integrated in terms of elliptic functions, as Euler discovered using his then newly invented theoryof elliptic integrals. Observe that the first equation of (6) reads

dm1

m2m3= (λ3 − λ2) dt, (7)



where m1,m2 and m3 are related by

λ1m21 + λ2m

22 + λ3m

23 = c1, m2

1 + m22 + m2

3 = c2.

Therefore, if λ2 �= λ3, we have

m2 = ±

√c2λ3 − c1 + (λ1 − λ3) m2

1

λ3 − λ2, m3 = ±

√c1 − c2λ2 + (λ2 − λ1)m2

1

λ3 − λ2.

Substituting these expressions into (7), we find after integration that the system (6) amounts to anelliptic integral

m1(t)∫

m1(0)

dm√

(m2 + a) (m2 + b)= ct,

with respect to the elliptic curve

C : w2 =(z2 + a

) (z2 + b

), (8)

with a =c2λ3 − c1

λ1 − λ3, b =

c1 − c2λ2

λ2 − λ1, c =

√(λ1 − λ3) (λ2 − λ1). Then the functions mi(t) can be ex-

pressed in terms of theta-functions of t, according to the classical inversion of abelian integrals.

We shall use the Lax representation of the equations of motion to show that the linearized Euler flowcan be realized on an elliptic curve isomorphic to the original elliptic curve (8). The solution to (5) hasthe form

M (t) = O (t) M (t) M (t) ,

where O (t) is one parameter sub-group of SO (3). So the hamiltonian flow (5) preserves the spectrum ofX and therefore its characteristic polynomial det (M − zI) = −z

(z2 + m2

1 + m22 + m2

3

). Unfortunately,

the spectrum of a 3× 3 skew- symmetric matrix provides only one piece of information; the conservationof energy does not appear as part of the spectral information. Therefore one is let to considering anotherformulation. The basic observation, due to Manakov [20], is that equation (5) is equivalent to the Laxequation

A = [A,B] ,

where A = M + αh, B = ΛM + βh, with a formal indeterminate h and

α =

⎛

⎜⎜⎜⎝

α1 0 0

0 α2 0

0 0 α3

⎞

⎟⎟⎟⎠

, β =

⎛

⎜⎜⎜⎝

α1 0 0

0 β2 0

0 0 β3

⎞

⎟⎟⎟⎠

,

λ1 =β3 − β2

α3 − α2, λ2 =

β1 − β3

α1 − α3, λ3 =

β2 − β1

α2 − α1,

and all αi distinct. The characteristic polynomial of A is

P (h, z) = det (A − zI)

= det (M + αh − zI)

=3∏

j=1

(αjh − z) +

⎛

⎝3∑

j=1

αjm2j

⎞

⎠h −

⎛

⎝3∑

j=1

m2j

⎞

⎠ z.


302 LESFARI

The spectrum of the matrix A = M + αh as a function of h ∈ C is time independent and is given by thezeroes of the polynomial P (h, z), thus defining an algebraic curve (spectral curve). Letting w = h/z, weobtain the following elliptic curve

z23∏

j=1

(αjw − 1) + 2H1w − 2H2 = 0,

which is shown to be isomorphic to the original elliptic curve. Finally, we have the

Theorem 7. The Euler rigid body motion is a completely integrable system and the linearizedflow can be realized on an elliptic curve.

2.2. The Geodesic Flow for a Left Invariant Metric on SO(4)

Consider the group SO(4) and its Lie algebra so(4) paired with itself, via the customary inner product

〈X,Y 〉 = −12

tr (X.Y ), where

X =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 −x3 x2 −x4

x3 0 −x1 −x5

−x2 x1 0 −x6

x4 x5 x6 0

⎞

⎟⎟⎟⎟⎟⎟⎠

∈ so(4).

A left invariant metric on SO(4) is defined by a non-singular symmetric linear map Λ: so(4) −→ so(4),X −→ Λ.X, and by the following inner product; given two vectors gX and gY in the tangent spaceSO(4) at the point g ∈ SO(4), 〈gX, gY 〉 =

⟨X,Λ−1.Y

⟩. Then the geodesic flow for this metric takes

the following commutator form (Euler-Arnold equations):

X = [X,Λ.X] , (9)

where

Λ.X =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 −λ3x3 λ2x2 −λ4x4

λ3x3 0 −λ1x1 −λ5x5

−λ2x2 λ1x1 0 −λ6x6

λ4x4 λ5x5 λ6x6 0

⎞

⎟⎟⎟⎟⎟⎟⎠

∈ so(4).

In view of the isomorphism between(R

6,∧), and (so (4) , [, ]) we write the system (9) as

x1 = (λ3 − λ2)x2x3 + (λ6 − λ5) x5x6, x2 = (λ1 − λ3) x1x3 + (λ4 − λ4)x4x6,

x3 = (λ2 − λ1)x1x2 + (λ5 − λ4) x4x5, x4 = (λ3 − λ5) x3x5 + (λ6 − λ2)x2x6,

x5 = (λ4 − λ3)x3x4 + (λ1 − λ6) x1x6, x6 = (λ2 − λ4) x2x4 + (λ5 − λ1)x1x5.

These equations can be written as a hamiltonian vector field

x(t) = J∂H

∂x, x ∈ R

6, (10)

with

H =12〈X,ΛX〉 =

12(λ1x

21 + λ2x

22 + · · · + λ6x

26

),



the hamiltonian and

J =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 −x3 x2 0 −x6 x5

x3 0 −x1 x6 0 −x4

−x2 x1 0 −x5 x4 0

0 −x6 x5 0 −x3 x2

x6 0 −x4 x3 0 −x1

−x5 x4 0 −x2 x1 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

∈ so(6).

We have detJ = 0, so m = 2n + k and m − k = rk J . Here m = 6 and rg J = 4, then n = k = 2. Thesystem (10) has beside the energy H1 = H , two trivial constants of motion:

H2 =12(x2

1 + x22 + · · · + x2

6

), H3 = x1x4 + x2x5 + x3x6.

Recall that H2 and H3 are called trivial invariants (or Casimir functions) because J∂H2

∂x= J

∂H3

∂x= 0.

In order that the hamiltonian system (10) be completely integrable, it is suffices to have one moreintegral, which we take of the form

H4 =12(μ1x

21 + μ2x

22 + · · · + μ6x

26

).

The four invariants must be functionally independent and in involution, so in particular

{H4,H3} =⟨

∂H4

∂x, J

∂H3

∂x

⟩

= 0,

i.e.,

((λ3 − λ2) μ1 + (λ1 − λ3)μ2 + (λ2 − λ1)μ3)x1x2x3

+ ((λ6 − λ5) μ1 + (λ1 − λ6) μ5 + (λ5 − λ1) μ6) x1x5x6

+ ((λ4 − λ6) μ2 + (λ6 − λ2) μ4 + (λ2 − λ4) μ6) x2x4x6

+ ((λ5 − λ4) μ3 + (λ3 − λ5) μ4 + (λ4 − λ3)μ5) x3x4x5 = 0.

Then

(λ3 − λ2)μ1 + (λ1 − λ3) μ2 + (λ2 − λ1) μ3 = 0, (λ6 − λ5) μ1 + (λ1 − λ6)μ5 + (λ5 − λ1)μ6 = 0,

(λ4 − λ6)μ2 + (λ6 − λ2) μ4 + (λ2 − λ4) μ6 = 0, (λ5 − λ4) μ3 + (λ3 − λ5)μ4 + (λ4 − λ3)μ5 = 0.

Put

A =

⎛

⎜⎜⎜⎜⎜⎜⎝

λ3 − λ2 λ1 − λ3 λ2 − λ1 0 0 0

λ6 − λ5 0 0 0 λ1 − λ6 λ5 − λ1

0 λ4 − λ6 0 λ6 − λ2 0 λ2 − λ4

0 0 λ5 − λ4 λ3 − λ5 λ4 − λ3 0

⎞

⎟⎟⎟⎟⎟⎟⎠

.

The number of solutions of this system is equal to the number of columns of the matrix A minus the rankof A. If rkA = 4, we have two solutions: μi = 1 lead to the invariant H2 and μi = λi lead to the invariant


304 LESFARI

H3. This is unacceptable. If rkA = 3, each four-order minor of A is singular. Now⎛

⎜⎜⎜⎜⎜⎜⎝

λ3 − λ2 λ1 − λ3 λ2 − λ1 0

λ6 − λ5 0 0 0

0 λ4 − λ6 0 λ6 − λ2

0 0 λ5 − λ4 λ3 − λ5

⎞

⎟⎟⎟⎟⎟⎟⎠

= − (λ6 − λ5) C,

⎛

⎜⎜⎜⎜⎜⎜⎝

λ1 − λ3 λ2 − λ1 0 0

0 0 0 λ1 − λ6

λ4 − λ6 0 λ6 − λ2 0

0 λ5 − λ4 λ3 − λ5 λ4 − λ3

⎞

⎟⎟⎟⎟⎟⎟⎠

= (λ1 − λ6) C,

⎛

⎜⎜⎜⎜⎜⎜⎝

λ2 − λ1 0 0 0

0 0 λ1 − λ6 λ5 − λ1

0 λ6 − λ2 0 λ2 − λ4

λ5 − λ4 λ3 − λ5 λ4 − λ3 0

⎞

⎟⎟⎟⎟⎟⎟⎠

= − (λ2 − λ1) C,

whereC ≡ λ1λ6λ4 + λ1λ2λ5 − λ1λ2λ4 + λ3λ6λ5 − λ3λ6λ4 − λ3λ2λ5

+ λ4λ2λ5 + λ4λ1λ3 − λ4λ1λ5 + λ6λ2λ3 − λ6λ2λ5 − λ1λ6λ3,

and it follows that the condition for which these minors are zero is C = 0. Notice that this relation holdsby cycling the indices: 1 � 4, 2 � 5, � 3 � 6. Under Manakov [20] conditions,

λ1 =β2 − β3

α2 − α3, λ2 =

β1 − β3

α1 − α3, λ3 =

β1 − β2

α1 − α2, (11)

λ4 =β1 − β4

α1 − α4, λ5 =

β2 − β4

α2 − α4, λ6 =

β3 − β4

α3 − α4,

where αi, βi ∈ C,∏

i<j (αi − βj) �= 0, equations (10) admits a Lax equation with an indeterminate h:

˙(︷︸︸︷X + αh) = [X + αh,ΛX + βh] , (12)

α =

⎛

⎜⎜⎜⎜⎜⎜⎝

α1 0 0 0

0 α2 0 0

0 0 α3 0

0 0 0 α4

⎞

⎟⎟⎟⎟⎟⎟⎠

, β =

⎛

⎜⎜⎜⎜⎜⎜⎝

β1 0 0 0

0 β2 0 0

0 0 β3 0

0 0 0 β4

⎞

⎟⎟⎟⎟⎟⎟⎠

.

�

X = [X,Λ.X] ⇔ (9) ,

[X,β] + [α,Λ.X] = 0 ⇔ (11) ,

[α, β] = 0 trivially satisfied for diagonal matrices.

The parameters μ1, . . . , μ6 can be parameterized (like λ1, . . . , λ6) by:

μ1 =γ2 − γ3

α2 − α3, μ2 =

γ1 − γ3

α1 − α3, μ3 =

γ1 − γ2

α1 − α2, μ4 =

γ1 − γ4

α1 − α4, μ5 =

γ2 − γ4

α2 − α4, μ6 =

γ3 − γ4

α3 − α4.



To use the method of isospectral deformations, consider the Kac-Moody extension (n = 4): L ={∑N−∞ Aih

i : Narbitrary ∈ Z, Ai ∈ gl(n, R)}

, of gl(n, R) with the bracket:[∑

Aihi,∑

Bjhj]

=∑

k

(∑i+j=k [Ai, Bj ]

)hk, and the ad-invariant form:

⟨∑Aih

i,∑

Bjhj⟩

=∑

i+j=−1〈Ai, Bj〉, where

〈, 〉 is the usual form defined on gl(n, R). Let K and N be respectively the ≥ 0 and < 0 powers of h

in L, then L = K + N , for the pairing defined above K = K⊥, N = N⊥, so that K = N ∗. The orbitsdescribed in this way come equipped with a symplectic structure with Poisson bracket {H1,H2} (α) =〈α, [∇K∗H1,∇K∗H2]〉, where α ∈ K∗ and ∇K∗H ∈ K. According to the Adler-Kostant-Symes theorem,the flow (12) is hamiltonian on an orbit through the point X + ah, X ∈ so(4) formed by the coadjointaction of the subgroup GN ⊂ SL (n) of lower triangular matrices on the dual Kac-Moody algebraN ∗ ≈ K⊥ = K. As a consequence, the coefficients of zihi appearing in curve:

Γ :{(z, h) ∈ C

2 : det (X + ah − zI) = 0}

, (13)

associated to the equation (12), are invariant of the system in involution for the symplectic structure ofthis orbit. Notice that

det(gXg−1

)= detX = (x1x4 + x2x5 + x3x6)

2 ,

tr(gXg−1

)2 = tr(gX2g−1

)= tr

(X2

)= −2

(x2

1 + x22 + · · · + x2

6

).

Also the complex flows generated by these invariants can be realized as straight lines on the abelianvariety defined by the periods of curve Γ. Explicitly, equation (13) looks as follows

Γ :4∏

i=1

(αih − z) + 2H4h2 − 2H1zh + 2H2z

2 + H23 = 0,

where H1(X) = c1, H2(X) = c2, H3(X) = 2H = c3, H4(X) = c4, with c1, c2, c3, c4 generic constants.Γ is a curve of genus 3 and it has a natural involution σ: Γ → Γ, (z, h) → (−z,−h). Therefore thejacobian variety Jac(Γ) of Γ splits up into an even and old part: the even part is an elliptic curve Γ0 = Γ/σand the odd part is a 2-dimensional abelian surface Prym(Γ/Γ0) called the Prym variety: Jac(Γ) = Γ0 +Prym(Γ/Γ0). The van Moerbeke-Mumford linearization method provides then an algebraic map fromthe complex affine variety

⋂4i=1 {Hi(X) = ci} ⊂ C

6 to the Jacobi variety Jac(Γ). By the antisymmetryof Γ, this map sends this variety to the Prym variety Prym(Γ/Γ0):

4⋂

i=1

{Hi(X) = ci} → Prym(Γ/Γ0), p →3∑

k=1

sk,

and the complex flows generated by the constants of the motion are straight lines on Prym(Γ/Γ0).Finally, we have the

Theorem 8. The geodesic flow (9) is a hamiltonian system with

H ≡ H1 =12(λ1x

21 + λ2x

22 + · · · + λ6x

26

),

the hamiltonian. It has two trivial invariants

H2 =12(x2

1 + x22 + · · · + x2

6

), H3 = x1x4 + x2x5 + x3x6.

Moreover, ifλ1λ6λ4 + λ1λ2λ5 − λ1λ2λ4 + λ3λ6λ5 − λ3λ6λ4 − λ3λ2λ5

+λ4λ2λ5 + λ4λ1λ3 − λ4λ1λ5 + λ6λ2λ3 − λ6λ2λ5 − λ1λ6λ3 = 0,

the system (9) has a fourth independent constant of the motion of the form

H4 =12(μ1x

21 + μ2x

22 + · · · + μ6x

26

).

Then the system (9) is completely integrable and can be linearized on the Prym varietyPrym(Γ/Γ0).


306 LESFARI

2.3. The Toda Lattice

The Toda lattice equations (discretized version of the Korteweg-de Vries equation3 ) motion of nparticles with exponential restoring forces are governed by the following hamiltonian

H =12

N∑

i=1

p2i +

N∑

i=1

eqi−qi+1, qN+1 = q1.

The hamiltonian equations can be written as follows

qi = pi, pi = −eqi−qi+1 + eqi−1−qi .

In term of the Flaschka’s variables [6]: ai = 12eqi−qi+1 , bi = −1

2pi, Toda’s equations take the followingform

ai = ai (bi+1 − bi) , bi = 2(a2i − a2

i−1), (14)

with bN+1 = b1 and a0 = aN . To show that the system (14) is completely integrable, one should find Nfirst integrals independent and in involution each other. From the second equation, we have

d

dt

N∑

i=1

bi =N∑

i=1

dbi

dt= 0,

and we normalize the bi’s by requiring that∑N

i=1 bi = 0. This is a first integral for the system. We furtherdefine N × N matrices A and B with

A =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

b1 a1 0 · · · aN

a1 b2...

...

0. . . . . . . . . 0

.... . . bN−1 aN−1

aN · · · 0 aN−1 bN

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, B =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 a1 · · · · · · −aN

−a1 0...

......

. . . . . . . . ....

.... . . . . . aN−1

aN · · · · · · −aN−1 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Then (14) is equivalent to the Lax equation

A = [B,A] .

From Theorem 3, we know that the quantities Ik =1ktrAk, 1 ≤ k ≤ N , are first integrals of motion: To

be more precise

Ik = tr(A.Ak−1) = tr([B,A].Ak−1) = tr(BAk − ABAk−1) = 0.

Notice that I1 is the first integral already know. Since these N first integrals are shown to be independentand in involution each other, the system (14) is thus completely integrable.

2.4. The Garnier Potential

Consider the hamiltonian

H =12(x2

1 + x22

)− 1

2(λ1y

21 + λ2y

22

)+

14(y21 + y2

2

)2, (15)

where λ1 and λ2 are constants. The corresponding system is given by

y1 = x1, x1 =(λ1 − y2

1 − y22

)y1, (16)

y2 = x2, x2 =(λ2 − y2

2 − y21

)y2.

3 In short K-dV equation: ∂u∂t

− 6u∂u∂x

+ ∂3u∂x3 = 0. This is an infinite-dimensional completely integrable system.



Theorem 9. The system (16) has the additional first integral

H2 =14

((x1y2 − x2y1)

2 −(λ2y

41 + λ1y

42

)− (λ1 + λ2) y2

1y22

)+

12(λ1λ2

(y21 + y2

2

)−(λ2x

21 + λ1x

22

))

and is completely integrable. The flows generated by H1 = H(15) and H2 are straight linemotions on the jacobian variety of a smooth genus two hyperelliptic curve H(17) associated to aLax equation.

Proof. We consider the Lax representation in the form A = [A,B], with the following ansatz for theLax operator

A =

⎛

⎝ U V

W −U

⎞

⎠ , B =

⎛

⎝ 0 1

R 0

⎞

⎠

where

V = −(h − λ1)(h − λ2)(

1 +12

(y21

h − λ1+

y22

h − λ2

))

,

U =12(h − λ1)(h − λ2)

(x1y1

h − λ1+

x2y2

h − λ2

)

,

W = (h − λ1)(h − λ2)(

12

(x2

1

h − λ1+

x22

h − λ2

)

− h +12(y2

1 + y22))

,

R = h − y21 − y2

2.

We form the curve in (z, h) space

P (h, z) = det (A − zI) = 0,

whose coefficients are functions of the phase space. Explicitly, this equation looks as follows

H : z2 = P5 (h) (17)

= (h − λ1)(h − λ2)(h3 − (λ1 + λ2)h2 + (λ1λ2 − H1)h − H2),

with H1 (15) the hamiltonian and a second quartic integral H2 of the form

H2 = −14(λ2y

41 + λ1y

42 + (λ1 + λ2)y2

1y22 − (x1y2 − x2y1)2) −

12(λ2x

21 + λ1x

22 − λ1λ2(y2

1 + y22)).

The functions H1 and H2 commute: {H1,H2} = 0 and the system (16) is completely integrable. Thecurve H determined by the fifth-order equation (17) is smooth, hyperelliptic and its genus is 2. Obviously,H is invariant under the hyperelliptic involution (h, z) � (h,−z). Using the van Moerbeke-Mumfordlinearization method, we show that the linearized flow could be realized on the jacobian variety Jac (H)of the genus 2 curve H. For generic c = (c1, c2) ∈ C

2 the affine variety defined by

Mc =2⋂

i=1

{x ∈ C

4 : Hi (x) = ci

},

is a smooth affine surface. According to the schema of [5], we introduce coordinates s1 and s2 on thesurface Mc, such that Mc (si) = 0, λ1 �= λ2, i.e.,

s1 + s2 =12(y21 + y2

2

)+ λ1 + λ2, s1s2 =

12(λ2y

21 + λ1y

22

)+ λ1λ2.

After some algebraic manipulations, we obtain the following equations for s1 and s2:

s1 = 2

√P5 (s1)

s1 − s2, s2 = 2

√P5 (s2)

s2 − s1,


308 LESFARI

where P5 (s) is defined by (17). These equations can be integrated by the abelian mapping

H −→ Jac (H) = C2/L, p −→

⎛

⎝

p∫

p0

ω1,

p∫

p0

ω2

⎞

⎠ ,

where the hyperelliptic curve H of genus two is given by the equation (17), L is the lattice generatedby the vectors n1 + Ωn2, (n1, n2) ∈ Z

2,Ω is the matrix of period of the curve H, (ω1, ω2) is a canonicalbasis of holomorphic differentials on H, i.e.,

ω1 =ds

√P5 (s)

, ω2 =sds

√P5 (s)

,

and p0 is a fixed point. This concludes the proof of the theorem.

2.5. The Coupled Nonlinear Schrodinger Equations

The system of two coupled nonlinear Schrodinger equations is given by

i∂a

∂z+

∂2a

∂t2+ Ω0a +

23

(|a|2 + |b|2

)a +

13(a2 + b2

)a = 0, (18)

i∂b

∂z+

∂2b

∂t2− Ω0b +

23

(|a|2 + |b|2

)b +

13(a2 + b2

)b = 0,

where a (z, t) and b (z, t) are functions of z and t, the bar “−” denotes the complex conjugation, “||”denotes the modulus and Ω0 is a constant. These equations play a significant role in mathematics, witha important number of physical applications. We seek solutions of (18) in the following form

a (z, t) = y1 (t) exp (iΩz) , b (z, t) = y2 (t) exp (iΩz) ,

where y1 (t) et y2 (t) are two functions and Ω is an arbitrary constant. Then we obtain the system

y1 +(y21 + y2

2

)y1 = (Ω − Ω0) y1, y2 +

(y21 + y2

2

)y2 = (Ω + Ω0) y2.

The latter coincides obviously with (16) for λ1 = Ω − Ω0 and λ2 = Ω + Ω0.

2.6. The Yang-Mills Equations

We consider the Yang-Mills system for a field with gauge group SU(2):

�jFjk =∂Fjk

∂τj+ [Aj , Fjk] = 0,

where Fjk, Aj ∈ TeSU(2), 1 ≤ j, k ≤ 4 and Fjk =∂Ak

∂τj− ∂Aj

∂τk+ [Aj , Ak]. The self-dual Yang-Mills

(SDYM) equations is an universal system for which some reductions include all classical tops fromEuler to Kowalewski (0 + 1-dimensions), K-dV, Nonlinear Schrodinger, Sine-Gordon, Toda latticeand N-waves equations (1 + 1-dimensions), KP and D-S equations (2 + 1-dimensions). In thecase of homogeneous double-component field, we have ∂jAk = 0, j �= 1, A1 = A2 = 0, A3 = n1U1 ∈su(2), A4 = n2U2 ∈ su(2) where ni are su(2)-generators (i.e., they satisfy commutation relations:n1 = [n2, [n1, n2]], n2 = [n1, [n2, n1]]). The system becomes

∂2U1

∂t2+ U1U

22 = 0,

∂2U2

∂t2+ U2U

21 = 0,

with t = τ1. By setting Uj = qj ,∂Uj

∂t= pj , j = 1, 2, Yang-Mills equations are reduced to hamiltonian

system

x = J∂H

∂x, x = (q1, q2, p1, p2)ᵀ, J =

⎛

⎝ O −I

I O

⎞

⎠ ,



with H =12(p2

1 + p22 + q2

1q22), the hamiltonian. The symplectic transformation p1 �

√2

2(p1 + p2), p2 �

√2

2(p1 − p2), q1 �

12( 4√

2)(q1 + iq2), q2 �12( 4√

2)(q1 − iq2), takes this hamiltonian into

H =12(p2

1 + p22) +

14q41 +

14q42 +

12q21q

22 ,

which coincides with (15) for λ1 = λ2 = 0.

3. ALGEBRAIC COMPLETE INTEGRABILITY

We give some results about abelian surfaces which will be used, as well as the basic techniquesto study two-dimensional algebraic completely integrable systems. Let M = C/Λ be a n-dimensionalabelian variety where Λ is the lattice generated by the 2n columns λ1, . . . , λ2n of the n × 2n periodmatrix Ω and let D be a divisor on M . Define L(D) = {fmeromorphic onM : (f) ≥ −D}, i.e., forD =

∑kjDj a function f ∈ L(D) has at worst a kj-fold pole along Dj . The divisor D is called ample

when a basis (f0, . . . , fN ) of L(kD) embeds M smoothly into PN for some k, via the map M → P

N ,p → [1 : f1(p) : . . . : fN (p)], then kD is called very ample. It is known that every positive divisor D onan irreducible abelian variety is ample and thus some multiple of D embeds M into P

N . By a theoremof Lefschetz, any k ≥ 3 will work. Moreover, there exists a complex basis of C

n such that the latticeexpressed in that basis is generated by the columns of the n × 2n period matrix

⎛

⎜⎜⎜⎝

δ1 0 |. . . | Z

0 δn |

⎞

⎟⎟⎟⎠

,

with Z = Z, ImZ > 0, δj ∈ N∗ and δj |δj+1. The integers δj which provide the so-called polarization of

the abelian variety M are then related to the divisor as follows:

dimL(D) = δ1 . . . δn. (19)

In the case of a 2-dimensional abelian varieties (surfaces), even more can be stated: the geometric genusg of a positive divisor D (containing possibly one or several curves) on a surface M is given by theadjunction formula

g(D) =KM .D + D.D

2+ 1, (20)

where KM is the canonical divisor on M , i.e., the zero-locus of a holomorphic 2-form, D.D denote thenumber of intersection points of D with a +D (where a +D is a small translation by a of D on M ), whereas the Riemann-Roch theorem for line bundles on a surface tells you that

χ(D) = pa(M) + 1 +12(D.D −DKM ), (21)

where pa (M) is the arithmetic genus of M and χ(D) the Euler characteristic of D. To study abeliansurfaces using Riemann surfaces on these surfaces, we recall that

χ(D) = dimH0(M,OM (D)) − dimH1(M,OM (D)) (22)

= dimL(D) − dimH1(M,Ω2(D ⊗ K∗M )), (Kodaira-Serre duality)

= dimL(D), (Kodaira vanishing theorem),

whenever D ⊗ K∗M defines a positive line bundle. However for abelian surfaces, KM is trivial and

pa(M) = −1; therefore combining relations (19), (20), (21) and (22),

χ (D) = dimL(D) =D.D

2= g (D) − 1 = δ1δ2.


310 LESFARI

A divisor D is called projectively normal, when the natural map L(D)⊗k → L(kD), is surjective, i.e.,every function of L(kD) can be written as a linear combination of k-fold products of functions of L(D).Not every very ample divisor D is projectively normal but if D is linearly equivalent to kD0 for k ≥ 3 forsome divisor D0, then D is projectively normal.

Now consider the exact sheaf sequence

0 −→ OCπ∗−→ OC −→ X −→ 0,

where C is a singular connected Riemann surface, C =∑

Cj the corresponding set of smooth Riemannsurfaces after desingularization and π: C → C the projection. The exactness of the sheaf sequence showsthat the Euler characteristic

X (O) = dimH0(O) − dim H1(O),

satisfy

X (OC) −X (OC) + X (X) = 0, (23)

where X (X) only accounts for the singular points p of C; X (Xp) is the dimension of the set ofholomorphic functions on the different branches around p taken separately, modulo the holomorphicfunctions on the Riemann surface C near that singular point. Consider the case of a planar singularity(in this paper, we will be concerned by a tacnode for which X (X) = 2, as well), i.e., the tangents to thebranches lie in a plane. If fj(x, y) = 0 denote the jth branch of C running through p with local parametersj , then

X (Xp) = dim ΠjC[[sj]]/C[[x, y]]

Πjfj(x, y).

So using (22) and Serre duality, we obtain X (OC) = 1 − g(C) and X (OC) = n −∑n

j=1 g(Cj). Also,replacing in the formula (23), gives

g(C) =n∑

j=1

g(Cj) + X (X) + 1 − n.

Finally, recall that a Kahler variety is a variety with a Kahler metric, i.e., a hermitian metric whoseassociated differential 2-form of type (1, 1) is closed. The complex torus C

2/lattice with the euclideanmetric

∑dzi ⊗ dzi is a Kahler variety and any compact complex variety that can be embedded in

projective space is also a Kahler variety. Now, a compact complex Kahler variety having as manyindependent meromorphic functions as its dimension is a projective variety.

Consider now hamiltonian problems of the form

XH : x = J∂H

∂x≡ f(x), x ∈ R

m, (24)

where H is the hamiltonian and J = J(x) is a skew-symmetric matrix with polynomial entries in x, for

which the corresponding Poisson bracket {Hi,Hj} =⟨

∂Hi

∂x, J

∂Hj

∂x

⟩

, satisfies the Jacobi identities.

The system (24) with polynomial right hand side will be called algebraic complete integrable (a.c.i.)when:

(i) The system possesses n + k independent polynomial invariants H1, . . . ,Hn+k of which k lead to

zero vector fields J∂Hn+i

∂x(x) = 0, 1 ≤ i ≤ k, the n remaining ones are in involution (i.e., {Hi,Hj} = 0)

and m = 2n + k. For most values of ci ∈ R, the invariant varietiesn+k⋂

i=1{x ∈ R

m : Hi = ci} are assumed

compact and connected. Then, according to the Arnold-Liouville theorem, there exists a diffeomorphismn+k⋂

i=1

{x ∈ Rm : Hi = ci} → R

n/Lattice,



and the solutions of the system (24) are straight lines motions on these tori.(ii) The invariant varieties, thought of as affine varieties in C

m can be completed into complexalgebraic tori, i.e.,

n+k⋂

i=1

{Hi = ci, x ∈ Cm} ∪ D = C

n/Lattice,

where Cn/Lattice is a complex algebraic torus (i.e., abelian variety) and D a divisor. Algebraic means

that the torus can be defined as an intersectionM⋂

i=1

{Pi(X0, . . . ,XN ) = 0} involving a large number of

homogeneous polynomials Pi. In the natural coordinates (t1, . . . , tn) of Cn/Lattice coming from C

n,the functions xi = xi(t1, . . . , tn) are meromorphic and (24) defines straight line motion on C

n/Lattice.Condition (i) means, in particular, there is an algebraic map (x1(t), . . . , xm(t)) → (μ1(t), . . . , μn(t))making the following sums linear in t:

n∑

i=1

μi(t)∫

μi(0)

ωj = djt, 1 ≤ j ≤ n, dj ∈ C,

where ω1, . . . , ωn denote holomorphic differentials on some algebraic curves.The existence of a coherent set of Laurent solutions:

xi =∞∑

j=0

x(j)i tj−ki , ki ∈ Z, some ki > 0, (25)

depending on dim (phase space) −1 = m− 1 free parameters is necessary and sufficient for a hamiltoniansystem with the right number of constants of motion to be a.c.i. So, if the hamiltonian flow (24) is a.c.i.,it means that the variables xi are meromorphic on the torus C

n/Lattice and by compactness they mustblow up along a codimension one subvariety (a divisor) D ⊂ C

n/Lattice. By the a.c.i. definition, theflow (24) is a straight line motion in C

n/Lattice and thus it must hit the divisor D in at least one place.Moreover through every point of D, there is a straight line motion and therefore a Laurent expansionaround that point of intersection. Hence the differential equations must admit Laurent expansions whichdepend on the n − 1 parameters defining D and the n + k constants ci defining the torus C

n/Lattice,the total count is therefore m − 1 = dim (phase space) − 1 parameters.

Assume now hamiltonian flows to be (weight)-homogeneous with a weight νi ∈ N, going with eachvariable xi, i.e.,

fi (αν1x1, . . . , ανmxm) = ανi+1fi (x1, . . . , xm) , ∀α ∈ C.

Observe that then the constants of the motion H can be chosen to be (weight)-homogeneous:

H (αν1x1, . . . , ανmxm) = αkH (x1, . . . , xm) , k ∈ Z.

If the flow is algebraically completely integrable, the differential equations (24) must admits Laurentseries solutions (25) depending on m − 1 free parameters. We must have ki = νi and coefficients in theseries must satisfy at the 0th step non-linear equations,

fi

(x

(0)1 , . . . , x(0)

m

)+ gix

(0)i = 0, 1 ≤ i ≤ m, (26)

and at the kth step, linear systems of equations:

(L − kI) z(k) =

⎧⎨

⎩

0 for k = 1

some polynomial in x(1), . . . , x(k−1) for k > 1,(27)

where

L = Jacobian map of (26) =∂f

∂z+ gI |z=z(0) .


312 LESFARI

If m − 1 free parameters are to appear in the Laurent series, they must either come from the non-linearequations (26) or from the eigenvalue problem (27), i.e., L must have at least m− 1 integer eigenvalues.These are much less conditions than expected, because of the fact that the homogeneity k of the constantH must be an eigenvalue of L Moreover the formal series solutions are convergent as a consequence ofthe majorant method. Next we assume that the divisor is very ample and in addition projectively normal.Consider a point p ∈ D, a chart Uj around p on the torus and a function yj in L(D) having a pole ofmaximal order at p. Then the vector (1/yj , y1/yj, . . . , yN/yj) provides a good system of coordinates inUj . Then taking the derivative with regard to one of the flows

(yi

yj

)

˙=yiyj − yiyj

y2j

, 1 ≤ j ≤ N,

are finite on Uj as well. Therefore, since y2j has a double pole along D, the numerator must also have a

double pole (at worst), i.e., yiyj − yiyj ∈ L(2D). Hence, when D is projectively normal, we have that(

yi

yj

)

˙=∑

k,l

ak,l

(yk

yj

)(yl

yj

)

,

i.e., the ratios yi/yj form a closed system of coordinates under differentiation. At the bad points, theconcept of projective normality play an important role: this enables one to show that yi/yj is a bona fideTaylor series starting from every point in a neighbourhood of the point in question.

To prove the algebraic complete integrability of a given hamiltonian system, the main steps of themethod are:

—The first step is to show the existence of the Laurent solutions, which requires an argumentprecisely every time k is an integer eigenvalue of L and therefore L − kI is not invertible.

—One shows the existence of the remaining constants of the motion in involution so as to reach thenumber n + k.

—For given c1, . . . , cm, the set

D ≡

⎧⎨

⎩

xi (t) = t−νi

(x

(0)i + x

(1)i t + x

(2)i t2 + · · ·

), 1 ≤ i ≤ m

Laurent solutions such that : Hj (xi (t)) = cj + Taylor part ,

⎫⎬

⎭

defines one or several n − 1 dimentional algebraic varieties (divisor) having the property thatn+k⋂

i=1

{Hi = ci, z ∈ Cm} ∪ D = a smooth compact, connected variety

with n commuting vector fields

independent at every point.

= a complex algebraic torus T n = Cn/Lattice.

The flows J∂Hk+i

∂z, . . . , J

∂Hk+n

∂zare straight line motions on T n.

From the divisor D, a lot of information can be obtained with regard to the periods and the action-angle variables.

3.1. A Five-Dimensional System

Consider the following system of five differential equations in the unknowns z1, . . . , z5:

z1 = 2z4, z3 = z2(3z1 + 8z22), (28)

z2 = z3, z4 = z21 + 4z1z

22 + z5, z5 = 2z1z4 + 4z2

2z4 − 2z1z2z3.

The following three quartics are constants of motion for this system

F1 =12z5 − z1z

22 +

12z23 − 1

4z21 − 2z4

2 , (29)



F2 = z25 − z2

1z5 + 4z1z2z3z4 − z21z2

3 +14z41 − 4z2

2z24 , F3 = z1z5 + z2

1z22 − z2

4 .

This system is completely integrable and the hamiltonian structure is defined by the Poisson bracket

{F,H} =⟨

∂F

∂z, J

∂H

∂z

⟩

=∑5

k,l=1 Jkl∂F

∂zk

∂H

∂zl, where

∂H

∂z=(

∂H

∂z1,∂H

∂z2,∂H

∂z3,∂H

∂z4,∂H

∂z5

), and

J =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 2z1 4z4

0 0 1 0 0

0 −1 0 0 −4z1z2

−2z1 0 0 0 2z5 − 8z1z22

−4z4 0 4z1z2 −2z5 + 8z1z22 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

is a skew-symmetric matrix for which the corresponding Poisson bracket satisfies the Jacobi identities.

The system (28) can be written as z = J∂H

∂z, z = (z1, z2, z3, z4, z5), where H = F1. The second flow

commuting with the first is regulated by the equations z = J∂F2

∂z, z = (z1, z2, z3, z4, z5). These vector

fields are in involution: {F1, F2} =⟨

∂F1

∂z, J

∂F2

∂z

⟩

= 0, and the remaining one is casimir: J∂F3

∂z= 0.

The invariant variety A defined by

A =2⋂

k=1

{z : Fk(z) = ck} ⊂ C5, (30)

is a smooth affine surface for generic values of (c1, c2, c3) ∈ C3. So, the question I address is how does

one find the compactification of A into an abelian surface? The idea of the direct proof we shall givehere is closely related to the geometric spirit of the (real) Arnold-Liouville theorem. Namely, a compactcomplex n-dimensional variety on which there exist n holomorphic commuting vector fields which areindependent at every point is analytically isomorphic to a n-dimensional complex torus C

n/Latticeand the complex flows generated by the vector fields are straight lines on this complex torus. Now,the main problem will be to complete A (30) into a non singular compact complex algebraic varietyA = A∪D in such a way that the vector fields XF1 and XF2 generated respectively by F1 and F2, extendholomorphically along a divisor D and remain independent there. If this is possible, A is an algebraiccomplex torus (an abelian variety) and the coordinates z1, . . . , z5 restricted to A are abelian functions.A naive guess would be to take the natural compactification A of A by projectivizing the equations:A =

⋂3k=1{Fk(Z) = ckZ

40} ⊂ P

5. Indeed, this can never work for a general reason: an abelian variety Aof dimension bigger or equal than two is never a complete intersection, that is it can never be described insome projective space P

n by n-dim A global polynomial homogeneous equations. In other words, if A isto be the affine part of an abelian surface, A must have a singularity somewhere along the locus at infinityA ∩ {Z0 = 0}. In fact, we shall show that the existence of meromorphic solutions to the differentialequations (28) depending on 4 free parameters can be used to manufacture the tori, without ever goingthrough the delicate procedure of blowing up and down. Information about the tori can then be gatheredfrom the divisor.

Theorem 10. The system (28) possesses Laurent series solutions which depend on 4 freeparameters: α, β, γ and θ. These meromorphic solutions restricted to the surface A(30) areparameterized by two copies C−1 and C1 of the same Riemann surface (32) of genus 7.

Proof. The first fact to observe is that if the system is to have Laurent solutions depending on 4 freeparameters α, β, γ, θ, the Laurent decomposition of such asymptotic solutions must have the followingform

z1 =1tα − 1

2α2 + βt − 1

16α(α3 + 4β

)t2 + γt3 + · · · , (31)


314 LESFARI

z2 =12t

ε − 14εα +

18εα2t − 1

32ε(−α3 + 12β

)t2 + θt3 + · · · ,

z3 = − 12t2

ε +18εα2 − 1

16ε(−α3 + 12β

)t + 3θt2 + · · · ,

z4 = − 12t2

α +12β − 1

16α(α3 + 4β

)t +

32γt2 + · · · ,

z5 =1

2t2α2 − 1

4t(α3 + 4β

)+

14α(α3 + 2β

)−(α2β − 2γ + 4εθα

)t + · · · ,

with ε = ±1. Using the majorant method, we can show that these series are convergent. Substitutingthe Laurent solutions (31) into (29): F1 = c1, F2 = c2 and F3 = c3, and equating the t0-terms yields

F1 =764

α4 − 18αβ − 5

2εθ = c1, F2 =

116

(4β − α3

) (4α2β − α5 + 64εθα − 32γ

)= c2,

F3 = − 132

α6 − β2 − 14α3β − 3εθα2 + 4αγ = c3.

Eliminating γ and θ from these equations, leads to an equation connecting the two remaining parametersα and β:

C : 64β3 − 16α3β2 − 4(α6 − 32α2c1 − 16c3

)β + α

(32c2 − 32α4c1 + α8 − 16α2c3

)= 0. (32)

The Laurent solutions restricted to the surface A (30) are thus parameterized by two copies C−1 andC1 of the same Riemann surface C(32). According to the Riemann-Hurwitz formula, the genus of theRiemann surface C is 7, which establishes the theorem.

In order to embed C into some projective space, one of the key underlying principles used is theKodaira embedding theorem, which states that a smooth complex manifold can be smoothly embeddedinto projective space P

N with the set of functions having a pole of order k along positive divisor onthe manifold, provided k is large enough; fortunately, for abelian varieties, k need not be larger than threeaccording to Lefshetz. These functions are easily constructed from the Laurent solutions (31) by lookingfor polynomials in the phase variables which in the expansions have at most a k-fold pole. The nature ofthe expansions and some algebraic proprieties of abelian varieties provide a recipe for when to terminateour search for such functions, thus making the procedure implementable. Precisely, we wish to find a setof polynomial functions {f0, . . . , fN}, of increasing degree in the original variables z1, . . . , z5 having theproperty that the embedding D of C1 + C−1 into P

N via those functions satisfies the relation: geometricgenus (D) ≡ g(D) = N + 2. A this point, it may be not so clear why D must really live on an abeliansurface. Let us say, for the moment, that the equations of the divisor D (i.e., the place where the solutionsblow up), as a Riemann surface traced on the abelian surface A (to be constructed in Theorem 12), mustbe understood as relations connecting the free parameters as they appear firstly in the expansions (31).This means that (32) must be understood as relations connecting α and β. Let

L(r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

polynomials f = f(z, . . . , z5)

of degree ≤ r, such that

f(z(t)) = t−1(z(0) + . . .),

with z(0) �= 0 on D

and with z(t) as in (4)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

/[Fk = ck, k = 1, 2, 3],

and let (f0, f1, . . . , fNr) be a basis of L(r). We look for r such that: g(D(r)) = Nr + 2,D(r) ⊂ PNr . We

shall show that it is unnecessary to go beyond r = 4.

Theorem 11. (a) The spaces L(r), nested according to weighted degree, are generated as follows

L(1) = {f0, f1, f2}, L(2) = L(1) ⊕ {f3, f4, f5, f6}, (33)

L(3) = L(2) ⊕ {f7, f8, f9, f10}, L(4) = L(3) ⊕ {f12, f13, f14, f15},



where f0 = 1, f1 = z1, f2 = z2, f3 = 2z5 − z21 , f4 = z3 + 2εz2

2 , f5 = z4 + εz1z2, f6 = [f1, f2] ,f7 = f1(f1 + 2εf4), f8 = f2(f1 + 2εf4), f9 = z4(f3 + 2εf6), f10 = z5(f3 + 2εf6), f11 = f5(f1 + 2εf4),f12 = f1f2(f3 + 2εf6), f13 = f4f5 + [f1, f4] , f14 = [f1, f3] + 2ε [f1, f6] , f15 = f3 − 2z5 + 4f2

4 , with[sj , sk] = sjsk − sj sk, the wronskien of sk and sj .

(b) L(4) provides an embedding of D(4) into projective space P15 and D(4) has genus 17.

Proof. (a) The proof of (a) is straightforward and can be done by inspection of the expansions (31).

(b) It turns out that neither L(1), nor L(2), nor L(3), yields a Riemann surface of the right genus; infact g(D(r)) �= dim L(r) + 1, r = 1, 2, 3. For instance, the embedding into P

2 via L(1) does not separatethe sheets, so we proceed to L(2) and the corresponding embedding into P

6 is unacceptable sinceg(D(2))− 2 > 6 and D(2) ⊂ P

6 �= Pg−2, which contradicts the fact that Nr = g(D(2))− 2. So we proceed

to L(3) and we consider the corresponding embedding into P10, according to the functions (f0, . . . , f10).

For finite values of α and β, dividing the vector (f0, . . . , f10) by f2 and taking the limit t → 0, to

yield [0 : 2εα : 1 : −ε(4β − α3) : −α : −εα2 :12(4β − α3) : εα3 :

12α2 :

14εα3(4β − α3) : −1

4εα4(4β −

α3)]. The point α = 0 require special attention. Indeed near α = 0, the parameter β behaves asfollows: β ∼ 0, i

√c3,−i

√c3. Thus near (α, β) = (0, 0), the corresponding point is mapped into the

point [0 : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0] in P10 which is independent of ε = ±1, whereas near the

point (α, β) = (0, i√

c3) (resp. (α, β) = (0,−i√

c3)) leads to two different points: [0 : 0 : 1 : −4εi√

c3 :0 : 0 : 2εi

√c3 : 0 : 0 : 0 : 0] (resp. [0 : 0 : 1 : 4εi

√c3 : 0 : 0 : −2εi

√c3 : 0 : 0 : 0 : 0]), according to the

sign of ε. The Riemann surface (31) has three points covering α = ∞, at which β behaves as fol-

lows: β ∼ −1279216

α3,1

432α3

(1333 − 1295i

√3),

1432

α3(1333 + 1295i

√3). Then by dividing the vector

(f0, . . . , f10) by f10, the corresponding point is mapped into the point [0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 1]in P

10. Thus, g(D(3)) − 2 > 10 and D(2) ⊂ P10 �= P

g−2, which contradicts the fact that Nr = g(D(3)) −2. Consider now the embedding D(4) into P

15 using the 16 functions f0, . . . , f15 of L(4)(33). It is easilyseen that these functions separate all points of the Riemann surface (except perhaps for the pointsat α = ∞ and α = β = 0): The Riemann surfaces C1 and C−1 are disjoint for finite values of α andβ except for α = β = 0; dividing the vector (f0, . . . , f15) by f2 and taking the limit t → 0, to yield

[0 : 2εα : 1 : −ε(4β − α3) : −α : −εα2 :12(4β − α3) : εα3 :

12α2 :

14εα3(4β − α3) : −1

4εα4(4β − α3) :

−12εα4 : −1

4α3(4β − α3) :

34α(4β − α3

): εα3

(4β − α3

): −2εα3]. As before, the point α = 0 requires

special attention and the parameter β behaves as follows: β ∼ 0, i√

c3,−i√

c3. Thus near (α, β) = (0, 0),the corresponding point is mapped into the point [0 : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0]in P

15 which is independent of ε = ±1, whereas near the point (α, β) = (0, i√

c3) (resp. (α, β) =(0,−i

√c3)) leads to two different points : [0 : 0 : 1 : −4εi

√c3 : 0 : 0 : 2εi

√c3 : 0 : 0 : 0 : 0 : 0 : 0 : 0 :

0 : 0] (resp. [0 : 0 : 1 : 4εi√

c3 : 0 : 0 : −2εi√

c3 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0]), according to the sign ofε. About the point α = ∞, it is appropriate to divide by f10; then the corresponding point is mappedinto the point [0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 1 : 0 : 0 : 0 : 0 : 0], in P

15 which is independent of ε. Thedivisor D(4) obtained in this way has genus 17 and D(4) ⊂ P

15 = Pg−2, as desired. This ends the proof of

the theorem.Let L = L(4) and D = D(4). Next we wish to construct a surface strip around D which will support

the commuting vector fields. In fact, D has a good chance to be very ample divisor on an abelian surface,still to be constructed.

Theorem 12. The variety A(30) generically is the affine part of an abelian surface A. Thereduced divisor at infinity A\A = C1 + C−1, consists of two copies C1 and C−1 of the same genus7 Riemann surface C(32). The system of differential equations (28) is algebraically completelyintegrable and the corresponding flows evolve on A.

Proof. We need to attaches the affine part of the intersection of the three invariants F1, F2, F3 so asto obtain a smooth compact connected surface in P

15. To be precise, the orbits of the vector field (28)running through D form a smooth surface Σ near D such that Σ\A ⊆ A and the variety A = A ∪ Σ is


316 LESFARI

smooth, compact and connected. Indeed, let ψ(t, p) = {z(t) = (z1(t), . . . , z5(t)) : t ∈ C, 0 < |t| < ε},be the orbit of the vector field (28) going through the point p ∈ A. Let Σp ⊂ P

15 be the surface elementformed by the divisor D and the orbits going through p, and set Σ ≡ ∪p∈DΣp. Consider the Riemannsurface D′ = H ∩ Σ where H ⊂ P

15 is a hyperplane transversal to the direction of the flow. If D′ issmooth, then using the implicit function theorem the surface Σ is smooth. But if D′ is singular at 0,then Σ would be singular along the trajectory (t−axis) which go immediately into the affine part A.Hence, A would be singular which is a contradiction because A is the fibre of a morphism from C

5

to C3 and so smooth for almost all the three constants of the motion ck. Next, let A be the projective

closure of A into P5, let Z = [Z0 : Z1 : . . . : Z5] ∈ P

5 and let I = A ∩ {Z0 = 0} be the locus at infinity.Consider the map A ⊆ P

5 → P15, Z → f(Z), where f = (f0, f1, . . . , f15) ∈ L(D) and let A = f(A).

In a neighbourhood V (p) ⊆ P15 of p, we have Σp = A and Σp\D ⊆ A. Otherwise there would exist

an element of surface Σ′p ⊆ A such that Σp ∩ Σ′

p = (t − axis), orbit ψ(t, p) = (t − axis)\ p ⊆ A, and

hence A would be singular along the t−axis which is impossible. Since the variety A ∩ {Z0 �= 0} isirreducible and since the generic hyperplane section Hgen. of A is also irreducible, all hyperplane sectionsare connected and hence I is also connected. Now, consider the graph Γf ⊆ P

5 × P15 of the map f ,

which is irreducible together with A. It follows from the irreducibility of I that a generic hyperplanesection Γf ∩{Hgen. ×P

15} is irreducible, hence the special hyperplane section Γf ∩ {{Z0 = 0}×P15} is

connected and therefore the projection map projP15{Γf ∩{{Z0 = 0}×P15}} = f(I) ≡ D, is connected.

Hence, the variety A ∪ Σ = A is compact, connected and embeds smoothly into P15 via f . We wish

to show that A is an abelian surface equipped with two everywhere independent commuting vectorfields. For doing that, let φτ1 and φτ2 be the flows corresponding to vector fields XF1 and XF2 . Thelatter are generated respectively by F1 and F2. For p ∈ D and for small ε > 0, φτ1(p),∀τ1, 0 < |τ1| <

ε, is well defined and φτ1(p) ∈ A\A. Then we may define φτ2 on A by φτ2(q) = φ−τ1φτ2φτ1(q), q ∈U(p) = φ−τ1(U(φτ1(p))), where U(p) is a neighbourhood of p. By commutativity one can see thatφτ2 is independent of τ1; φ−τ1−ε1φτ2φτ1+ε1(q) = φ−τ1φ−ε1φτ2φτ1φε1 = φ−τ1φτ2φτ1(q). We affirm thatφτ2(q) is holomorphic away from D. This because φτ2φτ1(q) is holomorphic away from D and thatφτ1 is holomorphic in U(p) and maps bi-holomorphically U(p) onto U(φτ1(p)). Now, since the flowsφτ1 and φτ2 are holomorphic and independent on D, we can show along the same lines as in theArnold-Liouville theorem [15] that A is a complex torus C

2/lattice and so in particular A is a Kahlervariety. And that will done, by considering the local diffeomorphism C

2 → A, (τ1, τ2) → φτ1φτ2(p), fora fixed origin p ∈ A. The additive subgroup {(τ1, τ2) ∈ C

2 : φτ1φτ2(p) = p} is a lattice of C2, hence

C2/lattice → A is a biholomorphic diffeomorphism and A is a Kahler variety with Kahler metric given

by dτ1 ⊗ dτ1 + dτ2 ⊗ dτ2. As mentioned in appendix A, a compact complex Kahler variety having therequired number as (its dimension) of independent meromorphic functions is a projective variety. In fact,here we have A ⊆ P

15. Thus A is both a projective variety and a complex torus C2/lattice and hence an

abelian surface as a consequence of Chow theorem. This completes the proof of the theorem.

Remark 3.1. (a) Note that the reflection σ on the affine variety A amounts to the flip σ:(z1, z2, z3, z4, z5) → (z1,−z2, z3,−z4, z5), changing the direction of the commuting vector fields.It can be extended to the (-Id)-involution about the origin of C

2 to the time flip (t1, t2) →(−t1,−t2) on A, where t1 and t2 are the time coordinates of each of the flows XF1 and XF2 . Theinvolution σ acts on the parameters of the Laurent solution (30) as follows σ : (t, α, β, γ, θ) −→(−t,−α,−β,−γ, θ), interchanges the Riemann surfaces Cε and the linear space L can be split intoa direct sum of even and odd functions. Geometrically, this involution interchanges C1 and C−1,i.e., C−1 = σC1.

(b) Consider on A the holomorphic 1-forms dt1 and dt2 defined by dti(XFj ) = δij , where XF1

and XF2 are the vector fields generated respectively by F1 and F2. Taking the differentials ofζ = 1/z1 and ξ = z1/z2 viewed as functions of t1 and t2, using the vector fields and the Laurent



series (31) and solving linearly for dt1 and dt2, we obtain the holomorphic differentials

ω1 = dt1|Cε =1�

(∂ξ

∂t2dζ − ∂ζ

∂t2dξ

) ∣∣∣∣Cε

=8

α (−4β + α3)dα,

ω2 = dt2|Cε =1�

(−∂ξ

∂t1dζ − ∂ζ

∂t1dξ

) ∣∣∣∣Cε

=2

(−4β + α3)2dα,

with Δ ≡ ∂ζ

∂t1

∂ξ

∂t2− ∂ζ

∂t2

∂ξ

∂t1. The zeroes of ω2 provide the points of tangency of the vector field XF1

to Cε. We haveω1

ω2=

4α

(−4β + α3

), and XF1 is tangent to Hε at the point covering α = ∞.

3.2. The Henon-Heiles System

The Henon-Heiles system

q1 =∂H

∂p1, q2 =

∂H

∂p2, p1 = −∂H

∂q1, p2 = −∂H

∂q2, (34)

with

H ≡ H1 =12(p21 + p2

2 + aq21 + bq2

2

)+ q2

1q2 + 6q32,

has another constant of motion

H2 = q41 + 4q2

1q22 − 4p1 (p1q2 − p2q1) + 4aq2

1q2 + (4a − b)(p21 + aq2

1

),

where a, b, are constant parameters and q1, q2, p1, p2 are canonical coordinates and momenta, re-spectively. First studied as a mathematical model to describe the chaotic motion of a test star inan axisymmetric galactic mean gravitational field this system is widely explored in other branchesof physics. It well-known from applications in stellar dynamics, statistical mechanics and quantummechanics. It provides a model for the oscillations of atoms in a three-atomic molecule. The system(34) possesses Laurent series solutions depending on 3 free parameters α, β, γ, namely

q1 =α

t+(

α3

12+

αA

2− αB

12

)

t + βt2 + q(4)1 t3 + q

(5)1 t4 + q

(6)1 t5 + · · · ,

q2 = − 1t2

+α2

12− B

12+(

α4

48+

α2A

10− α2B

60− B2

240

)

t2 +αβ

3t3 + γt4 + · · · ,

where p1 =.q1, p2 =

.q2 and

q(4)1 =

αAB

24− α5

72+

11α3B

720− 11α3A

120− αB2

720− αA2

8, q

(5)1 = −βα2

12+

βB

60− Aβ

10,

q(6)1 = −αγ

9− α7

15552− α5A

2160+

α5B

12960+

α3B2

25920+

α3A2

1440− α3AB

4320+

αAB2

1440

− αB3

19440− αA2B

288+

αA3

144.

Let D be the pole solutions restricted to the surface

Mc =2⋂

i=1

{x ≡ (q1, q2, p1, p2) ∈ C

4,Hi (x) = ci

},

to be precise D is the closure of the continuous components of the set of Laurent series solutions x (t)such that Hi (x (t)) = ci, 1 ≤ i ≤ 2, i.e., D = t0—coefficient of Mc. Thus we find an algebraic curvedefined by

D : β2 = P8(α), (35)


318 LESFARI

where

P8 (α) = − 715 552

α8 − 1432

(

5A − 1318

B

)

α6 − 136

(671

15 120B2 +

177

A2 − 9431260

BA

)

α4

− 136

(

4A3 − 12520

B3 − 136

A2B +29AB2 − 10

7c1

)

α2 +136

c2.

The curve D determined by an eight-order equation is smooth, hyperelliptic and its genus is 3. Moreover,the map

σ : D −→ D, (β, α) −→ (β,−α), (36)

is an involution on D and the quotient E = D/σ is an elliptic curve defined by

E : β2 = P4(ζ), (37)

where P4 (ζ) is the degree 4 polynomial in ζ = α2 obtained from (35). The hyperelliptic curve D is thusa 2-sheeted ramified covering of the elliptic curve E (37),

ρ : D −→ E , (β, α) −→ (β, ζ), (38)

ramified at the four points covering ζ = 0 and ∞. The affine surface Mc completes into an abelian surfaceMc, by adjoining the divisor D. The latter defines on Mc a polarization (1, 2). The divisor 2D is very ampleand the functions 1, y1, y2

1, y2, x1, x21 + y2

1y2, x2y1 − 2x1y2, x1x2 + 2Ay1y2 + 2y1y22 , embed Mc smoothly

into CP7 with polarization (2, 4). Then the system (34) is algebraically completely integrable and the

corresponding flow evolues on an abelian surface Mc = C2/lattice, where the lattice is generated by the

period matrix

⎛

⎝ 2 0 a c

0 4 c b

⎞

⎠, Im

⎛

⎝ a c

c b

⎞

⎠ > 0.

Theorem 13. The abelian surface Mc which completes the affine surface Mc is the dual Prymvariety Prym∗ (D/E) of the genus 3 hyperelliptic curve D (35) for the involution σ interchangingthe sheets of the double covering ρ (38) and the problem linearizes on this variety.

Proof. Let (a1, a2, a3, b1, b2, b3) be a canonical homology basis of D such that σ (a1) = a3, σ (b1) =b3, σ (a2) = −a2, σ (b2) = −b2, for the involution σ (36). As a basis of holomorphic differentials

ω0, ω1, ω2 on the curve D (35) we take the differentials ω1 =α2dα

β, ω2 =

dα

β, ω3 =

αdα

β, and obviously

σ∗(ω1) = −ω1, σ∗(ω2) = −ω2, σ

∗(ω3) = ω3. Recall that the Prym variety Prym (D/E) is a subabelianvariety of the Jacobi variety Jac(D) = Pic0(D) = H1(OD) / H1(D, Z) constructed from the doublecover ρ: the involution σ on D interchanging sheets, extends by linearity to a map σ: Jac(D) → Jac(D)and up to some points of order two, Jac(D) splits into an even part and an odd part: the even part is anelliptic curve (the quotient of D by σ, i.e., E (18)) and the odd part is a 2-dimensional abelian surfacePrym (D/E). We consider the period matrix Ω of Jac(D)

Ω =

⎛

⎜⎜⎜⎜⎜⎜⎝

∫

a1

ω1

∫

a2

ω1

∫

a3

ω1

∫

b1

ω1

∫

b2

ω1

∫

b3

ω1

∫

a1

ω2

∫

a2

ω2

∫

a3

ω2

∫

b1

ω2

∫

b2

ω2

∫

b3

ω2

∫

a1

ω3

∫

a2

ω3

∫

a3

ω3

∫

b1

ω3

∫

b2

ω3

∫

b3

ω3

⎞

⎟⎟⎟⎟⎟⎟⎠

.

Then,

Ω =

⎛

⎜⎜⎜⎜⎜⎜⎝

∫

a1

ω1

∫

a2

ω1 −∫

a1

ω1

∫

b1

ω1

∫

b2

ω1 −∫

b1

ω1

∫

a1

ω2

∫

a2

ω2 −∫

a1

ω2

∫

b1

ω2

∫

b2

ω2 −∫

b1

ω2

∫

a1

ω3 0∫

a1

ω3

∫

b1

ω3 0∫

b1

ω3

⎞

⎟⎟⎟⎟⎟⎟⎠

,



and therefore the period matrices of Jac(E)(i.e., E), Prym(D/E) and Prym∗(D/E) are respectivelyΔ = (

∫a1

ω3

∫b1

ω3),

Γ =

⎛

⎜⎜⎝

2∫

a1

ω1

∫

a2

ω1 2∫

b1

ω1

∫

b2

ω1

2∫

a1

ω2

∫

a2

ω2 2∫

b1

ω2

∫

b2

ω2

⎞

⎟⎟⎠ ,

and

Γ∗ =

⎛

⎜⎜⎝

∫

a1

ω1

∫

a2

ω1

∫

b1

ω1

∫

b2

ω1

∫

a1

ω2

∫

a2

ω2

∫

b1

ω2

∫

b2

ω2

⎞

⎟⎟⎠ .

Let LΩ =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

∑3i=1 mi

∫ai

⎛

⎜⎜⎜⎝

ω1

ω2

ω3

⎞

⎟⎟⎟⎠

+ ni

∫bi

⎛

⎜⎜⎜⎝

ω1

ω2

ω3

⎞

⎟⎟⎟⎠

: mi, ni ∈ Z

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

, be the period lattice associated to Ω. Let

us denote also by LΔ, the period lattice associated Δ. We have the following diagram

0

↓

E D

↓ ϕ∗ ↙ ↓ ϕ

0 −→ ker Nϕ −→ Prym(D/E ⊕ E = Jac(D)Nϕ−→ E −→ 0

↘ τ ↓

Mc = Mc ∪ 2D � C2/lattice

↓

0

The polarization map τ : Prym(D/E) −→ Mc = Prym∗(D/E), has kernel (ϕ∗E) � Z2 ×Z2 and the in-duced polarization on Prym(D/E) is of type (1, 2). Let Mc → C

2/LΛ : p �∫ pp0

(dt1dt2

), be the uniformiz-

ing map where dt1, dt2 are two differentials on Mc corresponding to the flows generated respectively byH1, H2 such that: dt1|D = ω1 and dt2|D = ω2,

LΛ =

⎧⎪⎪⎨

⎪⎪⎩

4∑

k=1

nk

⎛

⎜⎜⎝

∫

νk

dt1

∫

νk

dt2

⎞

⎟⎟⎠ : nk ∈ Z

⎫⎪⎪⎬

⎪⎪⎭

,

is the lattice associated to the period matrix

Λ =

⎛

⎜⎜⎝

∫

ν1

dt1∫

ν2

dt1∫

ν4

dt1∫

ν4

dt1

∫

ν1

dt2∫

ν2

dt2∫

ν3

dt2∫

ν4

dt2

⎞

⎟⎟⎠ ,

and (ν1, ν2, ν3, ν4) is a basis of H1(Mc, Z). By the Lefschetz theorem on hyperplane section [9], the mapH1(D, Z) −→ H1(Mc, Z) induced by the inclusion D ↪→ Mc is surjective and consequently we can find


320 LESFARI

4 cycles ν1, ν2, ν3, ν4 on the curve D such that

Λ =

⎛

⎜⎜⎝

∫

ν1

ω1

∫

ν2

ω1

∫

ν4

ω1

∫

ν4

ω1

∫

ν1

ω2

∫

ν2

ω2

∫

ν3

ω2

∫

ν4

ω2

⎞

⎟⎟⎠ ,

and LΛ =

⎧⎨

⎩

∑4k=1 nk

⎛

⎝

∫νk

ω1∫νk

ω2

⎞

⎠ : nk ∈ Z

⎫⎬

⎭. The cycles ν1, ν2, ν3, ν4 in D which we look for are

a1, b1, a2, b2 and they generate H1(Mc, Z) such that

Λ =

⎛

⎜⎜⎝

∫

a1

ω1

∫

b1

ω1

∫

a2

ω1

∫

b2

ω1

∫

a1

ω2

∫

b1

ω2

∫

a2

ω2

∫

b2

ω2

⎞

⎟⎟⎠ ,

is a Riemann matrix. We show that Λ = Γ∗, i.e., the period matrix of Prym∗(D/E) dual of Prym(D/E).Consequently Mc and Prym∗(D/E) are two abelian varieties analytically isomorphic to the samecomplex torus C

2/LΛ. By Chow’s theorem [9], Ac and Prym∗(D/E) are then algebraically isomorphic.

3.3. The Kowalewski Rigid Body Motion

The motion for the Kowalewski’s top is governed by the equations.m = m ∧ λm + γ ∧ l,

.γ = γ ∧ λm, (39)

where m,γ and l denote respectively the angular momentum, the directional cosine of the z-axis (fixed inspace), the center of gravity which after some rescaling and normalization may be taken as l = (1, 0, 0)and λm = (m1/2,m2/2,m3/2). The system (39) can be written

.m1 = m2 m3,

.γ1 = 2m3γ2 − m2γ3, (40)

.m2 = −m1 m3 + 2γ3,

.γ2 = m1γ3 − 2m3γ1,

.m3 = −2γ2,

.γ3 = m2γ1 − m1γ2,

with constants of motion

H1 =12(m2

1 + m22

)+ m2

3 + 2γ1 = c1, (41)

H2 = m1γ1 + m2γ2 + m3γ3 = c2, H3 = γ21 + γ2

2 + γ23 = c3 = 1,

H4 =

((m1 + im2

2

)2

− (γ1 + iγ2)

)((m1 − im2

2

)2

− (γ1 − iγ2)

)

= c4.

The system (40) admits two distinct families of Laurent series solutions:

m1 (t) =

⎧⎪⎨

⎪⎩

α1

t+ i

(α2

1 − 2)α2 + ◦ (t) ,

α1

t− i

(α2

1 − 2)α2 + ◦ (t) ,

γ1 (t) =

⎧⎪⎨

⎪⎩

12t2

+ ◦ (t) ,

12t2

+ ◦ (t) ,

m2 (t) =

⎧⎪⎨

⎪⎩

iα1

t− α2

1α2 + ◦ (t) ,

−iα1

t− α2

1α2 + ◦ (t) ,

γ2 (t) =

⎧⎪⎨

⎪⎩

i

2t2+ ◦ (t) ,

−i

2t2+ ◦ (t) ,



m3 (t) =

⎧⎪⎨

⎪⎩

i

t+ α1α2 + ◦ (t) ,

−i

t+ α1α2 + ◦ (t) ,

γ3 (t) =

⎧⎪⎨

⎪⎩

α2

t+ ◦ (t) ,

α2

t+ ◦ (t) ,

which depend on 5 free parameters α1, . . . , α5. By substituting these series in the constants of the motionHi (41), one eliminates three parameters linearly, leading to algebraic relation between the two remainingparameters, which is nothing but the equation of the divisor D along which the mi, γi blow up. Sincethe system (40) admits two families of Laurent solutions, then D is a set of two isomorphic curves ofgenus 3, D = D1 + D−1:

Dε : P (α1, α2) =(α2

1 − 1) ((

α21 − 1

)α2

2 − P (α2))

+ c4 = 0, (42)

where P (α2) = c1α22 − 2εc2α2 − 1 and ε = ±1. Each of the curve Dε is a 2− 1 ramified cover (α1, α2, β)

of elliptic curves D0ε :

D0ε : β2 = P 2 (α2) − 4c4α

42, (43)

ramified at the 4 points α1 = 0 covering the 4 roots of P (α2) = 0. It was shown [12] that each divisor Dε

is ample and defines a polarization (1, 2), whereas the divisor D, of geometric genus 9, is very ample anddefines a polarization (2, 4). The affine surface Mc =

⋂4i=1 {Hi = ci} ⊂ C

6, defined by putting the fourinvariants (41) of the Kowalewski flow (40) equal to generic constants, is the affine part of an abeliansurface Mc with

Mc\Mc = D = one genus 9 curve consisting of two genus 3

curves Dε (42) intersecting in 4 points. Each

Dε is a double cover of an elliptic curve D0ε (43)

ramified at 4 points.

Moreover, the Hamiltonian flows generated by the vector fields XH1 and XH4 are straight lines onMc. The 8 functions 1, f1 = m1, f2 = m2, f3 = m3, f4 = γ3, f5 = f2

1 + f22 , f6 = 4f1f4 − f3f5, f7 =

(f2γ1 − f1γ2) f3 + 2f4γ2, form a basis of the vector space of meromorphic functions on Mc with at worsta simple pole along D Moreover, the map

Mc � C2/Lattice → CP

7, (t1, t2) → [(1, f1 (t1, t2) , . . . , f7 (t1, t2))] ,

is an embedding of Mc into CP7. Following the method (Theorem 13), we obtain the following theorem:

Theorem 14. The tori Mc can be identified as Mc = Prym∗(Dε/D0ε), i.e., dual of Prym(Dε/D0

ε)and the problem linearizes on this Prym variety.

3.4. Kirchhoff’s Equations of Motion of a Solid in an Ideal Fluid

The Kirchhoff’s equations of motion of a solid in an ideal fluid have the form

p1 = p2∂H

∂l3− p3

∂H

∂l2, l1 = p2

∂H

∂p3− p3

∂H

∂p2+ l2

∂H

∂l3− l3

∂H

∂l2, (44)

p2 = p3∂H

∂l1− p1

∂H

∂l3, l2 = p3

∂H

∂p1− p1

∂H

∂p3+ l3

∂H

∂l1− l1

∂H

∂l3,

p3 = p1∂H

∂l2− p2

∂H

∂l1, l3 = p1

∂H

∂p2− p2

∂H

∂p1+ l1

∂H

∂l2− l2

∂H

∂l1,

where (p1, p2, p3) is the velocity of a point fixed relatively to the solid, (l1, l2, l3) the angular velocityof the body expressed with regard to a frame of reference also fixed relatively to the solid and H is thehamiltonian. These equations can be regarded as the equations of the geodesics of the right-invariantmetric on the group E (3) = SO (3)× R

3 of motions of 3-dimensional euclidean space R3, generated by


322 LESFARI

rotations and translations. Hence the motion has the trivial coadjoint orbit invariants 〈p, p〉 and 〈p, l〉.As it turns out, this is a special case of a more general system of equations written as

x = x ∧ ∂H

∂x+ y ∧ ∂H

∂y, y = y ∧ ∂H

∂x+ x ∧ ∂H

∂y,

where x = (x1, x2, x3) ∈ R3 et y = (y1, y2, y3) ∈ R

3. The first set can be obtained from the second byputting (x, y) = (l, p/ε) and letting ε → 0. The latter set of equations is the geodesic flow on SO(4) fora left invariant metric defined by the quadratic form H . In Clebsch’s case, equations (44) have the fourinvariants:

H1 = H =12(a1p

21 + a2p

22 + a3p

23 + b1l

21 + b2l

22 + b3l

23

), H2 = p2

1 + p22 + p2

3,

H3 = p1l1 + p2l2 + p3l3, H4 =12(b1p

21 + b2p

22 + b3p

23 + �

(l21 + l22 + l23

)),

witha2 − a3

b1+

a3 − a1

b2+

a1 − a2

b3= 0, and the constant � satisfies the conditions � =

b1 (b2 − b3)a2 − a3

=

b2 (b3 − b1)a3 − a1

=b3 (b1 − b2)

a1 − a2. The system (44) can be written in the form (11) with m = 6; to be precise

x = f (x) ≡ J∂H

∂x, x = (p1, p2, p3, l1, l2, l3)ᵀ, (45)

where

J =

⎛

⎝ O P

P L

⎞

⎠ , P =

⎛

⎜⎜⎜⎝

0 −p3 p2

p3 0 −p1

−p2 p1 0

⎞

⎟⎟⎟⎠

, L =

⎛

⎜⎜⎜⎝

0 −l3 l2

l3 0 −l1

−l2 p1 0

⎞

⎟⎟⎟⎠

.

Consider points at infinity which are limit points of trajectories of the flow. In fact, there is a Laurentdecomposition of such asymptotic solutions,

x (t) = t−1(x(0) + x(1)t + x(2)t2 + . . .

), (46)

which depend on dim(phase space) − 1 = 5 free parameters. Putting (46) into (45), solving inductivelyfor the x(k), one finds at the 0th step a non-linear equation, x(0) + f(x(0)) = 0, and at the kth step, alinear system of equations,

(L − kI)x(k) =

⎧⎨

⎩

0 for k = 1

quadratic polynomial in x(1), . . . , x(k) for k ≥ 1,

where L denotes the jacobian map of the non-linear equation above. One parameter appear at the0th step, i.e., in the resolution of the non-linear equation and the 4 remaining ones at the kth step,k = 1, . . . , 4. Taking into account only solutions trajectories lying on the invariant surface Mc =4⋂

i=1{Hi (x) = ci} ⊂ C

6, we obtain one-parameter families which are parameterized by a curve:

D : θ2 + c1β2γ2 + c2α

2γ2 + c3α2β2 + c4αβγ = 0, (47)

where θ is an arbitrary parameter and where α = x(0)4 , β = x

(0)5 , γ = x

(0)6 parameterizes the elliptic curve

E : β2 = d21α

2 − 1, γ2 = d22α

2 + 1, (48)

with d1, d2 such that: d21 + d2

2 + 1 = 0. The curve D is a 2-sheeted ramified covering of the elliptic curve E .The branch points are defined by the 16 zeroes of c1β

2γ2 + c2α2γ2 + c3α

2β2 + c4αβγ on E . The curve Dis unramified at infinity and by Hurwitz’s formula, the genus of D is 9. Upon putting ζ ≡ α2, the curve Dcan also be seen as a 4-sheeted unramified covering of the following curve of genus 3:

C :(θ2 + c1β

2γ2 +(c2γ

2 + c3β2)ζ)2 − c2

4ζβ2γ2 = 0.



Moreover, the map τ : C → C, (θ, ζ) → (−θ, ζ), is an involution on C and the quotient C0 = C/τ is anelliptic curve defined by

C0 : η2 = c24ζ(d21d

22ζ

2 +(d21 − d2

2

)ζ − 1

).

The curve C is a double ramified covering of C0, C → C0, (θ, η, ζ) → (η, ζ),

C :

⎧⎨

⎩

θ2 = −c1β2γ2 −

(c2γ

2 + c3β2)ζ + η

η2 = c24ζ(d21d

22ζ

2 +(d21 − d2

2

)ζ − 1

).

Let (a1, a2, a3, b1, b2, b3) be a canonical homology basis of C such that τ (a1) = a3, τ (b1) = b3, τ (a2) =−a2 and τ (b2) = −b2 for the involution τ . Using the Poincare residu map, we show that

ω0 =dζ

η, ω1 =

ζdζ

θη, ω2 =

dζ

θη,

form a basis of holomorphic differentials on C and τ∗ (ω0) = ω0, τ∗ (ωk) = −ωk (k = 1, 2). The flow

evolues on an abelian surface Mc ⊆ CP7 of period matrix

⎛

⎝ 2 0 a c

0 4 c b

⎞

⎠ , Im

⎛

⎝ a c

c b

⎞

⎠ > 0. Following

the method (Theorem 13), we obtain

Theorem 15. The abelian surface Mc can be identified as Prym(C/C0). More precisely

4⋂

i=1

{x ∈ C

6,Hi (x) = ci

}= Prym(C/C0)\D,

where D is a genus 9 curve (47), which is a ramified cover of an elliptic curve E (48) with 16branch points.

4. GENERALIZED ALGEBRAIC COMPLETELY INTEGRABLE SYSTEMS

Some others integrable systems appear as coverings of algebraic completely integrable systems. Themanifolds invariant by the complex flows are coverings of abelian varieties and these systems are calledalgebraic completely integrable in the generalized sense.

Consider the case F3 = 0, (see Section 3.1) and the following change of variables

z1 = q21, z2 = q2, z3 = p2, z4 = p1q1, z5 = p2

1 − q21q

22 .

Substituting this into the constants of motion F1, F2, F3 leads obviously to the relations

H1 =12p21 −

32q21q

22 +

12p22 −

14q41 − 2q4

2 , (49)

H2 = p41 − 6q2

1q22p

21 + q4

1q42 − q4

1p21 + q6

1q22 + 4q3

1q2p1p2 − q41p

22 +

14q81,

whereas the last constant leads to an identity. Using the differential equations (28) combined with thetransformation above leads to the system of differential equations

q1 = q1

(q21 + 3q2

2

), q1 = q2

(3q2

1 + 8q22

). (50)

The last equation (28) for z5 leads to an identity. Thus, we obtain the potential constructed by Ramani,Dorozzi and Grammaticos [7, 25]. Evidently, the functions H1 and H2 commute: {H1,H2} = 0. Thesystem (50) is weight-homogeneous with q1, q2 having weight 1 and p1, p2 weight 2, so that H1 and H2

have weight 4 and 8 respectively. When one examines all possible singularities, one finds that it possiblefor the variable q1 to contain square root terms of the type t1/2, which are strictly not allowed by thePainleve test (i.e., the general solutions have no movable singularities other than poles). However, these


324 LESFARI

terms are trivially removed by introducing the variables z1, . . . , z5 which restores the Painleve propertyto the system. Let B be the affine variety defined by

B =2⋂

k=1

{z ∈ C4 : Hk(z) = bk}, (51)

where (b1, b2) ∈ C2.

Theorem 16. (a) The system (50) admits Laurent solutions in t1/2, depending on 3 freeparameters: u, v and w. These solutions restricted to the surface B(51) are parameterized bytwo copies Γ1 and Γ−1 of the same Riemann surface of genus 16.

(b) The invariant surface B (51) can be completed as a cyclic double cover B of the abeliansurface A, ramified along the divisor C1 + C−1. The system (50) is algebraic complete integrablein the generalized sense. Moreover, B is smooth except at the point lying over the singularity(of type A3) of C1 + C−1 and the resolution B of B is a surface of general type with invariants:X (B) = 1 and pg(B) = 2.

Proof. (a) The system (50) possesses 3-dimensional family of Laurent solutions (principal balances)depending on three free parameters u, v and w. There are precisely two such families, labeled by ε = ±1,and they are explicitly given as follows

q1 =1√t

(

u − 14u3t + vt2 − 5

128u7t3 +

18u

(34u3v − 7

256u8 + 3εw

)

t4 + · · ·)

, (52)

q2 =1t

(12ε − 1

4εu2t +

18εu4t2 +

14εu

(132

u5 − 3v)

t3 + wt4 + · · ·)

,

p1 =1

2t√

t

(

−u − 14u3t + 3vt2 − 25

128t3u7 +

78u

(34u3v − 7

256u8 + 3εw

)

t4 + · · ·)

,

p2 =1t2

(

−12ε +

18εu4t2 +

12εu

(132

u5 − 3v)

t3 + 3wt4 + · · ·)

.

These formal series solutions are convergent as a consequence of the majorant method. By substitutingthese series in the constants of the motion H1 = b1 and H2 = b2, one eliminates the parameter w linearly,leading to an equation connecting the two remaining parameters u and v:

Γ :654

uv3 +9364

u6v2 +3

8192(−9829u8 + 26112H1

)u3v (53)

− 10 29965 536

u16 − 123256

H1u8 + H2 +

1536 298 73152

= 0.

According to Hurwitz’ formula, this defines a Riemann surface Γ of genus 16. The Laurent solutionsrestricted to the surface B(51) are thus parameterized by two copies Γ−1 and Γ1 of the same Riemannsurface Γ.

(b) The morphism ϕ: B −→ A, (q1, q2, p1, p2) −→ (z1, z2, z3, z4, z5), maps the vector field (50) intoan algebraic completely integrable system (1) in five unknowns and the affine variety B(51) onto theaffine part A(30) of an abelian variety A with A\A = C1 + C−1. Observe that ϕ is an unramified cover.The Riemann surface Γ(53) play an important role in the construction of a compactification B of B.Let us denote by G a cyclic group of two elements {−1, 1} on V j

ε = U jε × {τ ∈ C : 0 < |τ | < δ}, where

τ = t1/2 and U jε is an affine chart of Γε for which the Laurent solutions (52) are defined. The action of

G is defined by (−1) ◦ (u, v, τ) = (−u,−v,−τ) and is without fixed points in V jε . So we can identify

the quotient V jε /G with the image of the smooth map hj

ε: V jε → B defined by the expansions (10).

We have (−1, 1).(u, v, τ) = (−u,−v, τ) and (1,−1).(u, v, τ) = (u, v,−τ), i.e., G × G acts separatelyon each coordinate. Thus, identifying V j

ε /G2 with the image of ϕ ◦ hjε in A. Note that Bj

ε = V jε /G is

smooth (except for a finite number of points) and the coherence of the Bjε follows from the coherence



of V jε and the action of G. Now by taking B and by gluing on various varieties Bj

ε\{some points}, weobtain a smooth complex manifold B which is a double cover of the abelian variety A (constructed inproposition 2.3) ramified along C1 + C−1, and therefore can be completed to an algebraic cyclic cover ofA. To see what happens to the missing points, we must investigate the image of Γ × {0} in ∪Bj

ε . Thequotient Γ × {0}/G is birationally equivalent to the Riemann surface Υ of genus 7:

Υ :654

y3 +9364

x3y2 +3

8192(−9829x4 + 26112b1

)x2y

+ x

(

−10 29965 536

x8 − 123256

b1x4 + b2 +

1536 298 73152

)

= 0,

where y = uv, x = u2. The Riemann surface Υ is birationally equivalent to C. The only points of Υ fixedunder (u, v) → (−u,−v) are the points at ∞, which correspond to the ramification points of the map

Γ × {0} 2−1→ Υ : (u, v) → (x, y) and coincides with the points at ∞ of the Riemann surface C. Then thevariety B constructed above is birationally equivalent to the compactification B of the generic invariantsurface B. So B is a cyclic double cover of the abelian surface A ramified along the divisor C1 + C−1,where C1 and C−1 have two points in commune at which they are tangent to each other. It follows thatThe system (8) is algebraic complete integrable in the generalized sense. Moreover, B is smooth exceptat the point lying over the singularity (of type A3) of C1 + C−1. In term of an appropriate local holomorphiccoordinate system (X,Y,Z), the local analytic equation about this singularity is X4 + Y 2 + Z2 = 0.Now, let B be the resolution of singularities of B, X (B) be the Euler characteristic of B and pg(B) thegeometric genus of B. Then B is a surface of general type with invariants: X (B) = 1 and pg(B) = 2.This concludes the proof of the theorem.

Remark 4.1. The asymptotic solution (52) can be read off from (31) and the change of variable:q1 =

√z1, q2 = z2, p1 = z4/q1, p2 = z3. The function z1 has a simple pole along the divisor C1 + C−1

and a double zero along a Riemann surface of genus 7 defining a double cover of A ramified alongC1 + C−1.

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