## Abstract

We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler–van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems.

## 1. Introduction

The purpose of this paper is to give a short survey on algebraic integrability, with the purpose of making the ideas and techniques accessible to a larger community. We present the general theory, giving at each step an example that is simple but non-trivial, highlighting the concept that has been introduced.

The paper is divided in two parts (§§2 and 3). In §2 we first recall the notion of a Poisson manifold and then the main examples, namely symplectic manifolds and the dual of a Lie algebra (§2*a*). We introduce in §2*b* the notion of an integrable system on a Poisson manifold and formulate the classical Liouville theorem, adapted to the case of Poisson manifolds. The major tool for constructing integrable systems is the Adler–Kostant–Symes theorem, given in §2*c*.

Section 3 deals with algebraic integrability. We mainly focus on algebraic complete integrability in the sense of Adler–van Moerbeke (a.c.i.). We present in detail the example of the periodic 5-particle Kac–van Moerbeke lattice in §3*b*. This section is the most technical part of the paper, aimed at showing, on an example, how algebraic integrability can be proven, when only the functions in involution are given (no Lax equation or geometric data are given). It can be skipped on first reading; it is included because it contains many useful ideas and illustrates almost all difficulties that are generally encountered when proving algebraic integrability. We discuss in §3*c* Lax equations with a parameter, a major source of a.c.i. systems. We finish the paper with some comments on the general notion of algebraic integrability and illustrate the latter with two examples.

## 2. Integrable systems on Poisson manifolds

The phase space of the simplest mechanical systems has the structure of a symplectic manifold: a smooth manifold *M* which is equipped with a closed, non-degenerate two-form, which allows one to identify the tangent and cotangent bundles to *M* and hence to associate to any one-form on *M* a vector field on *M*. Using the energy (function) *H* of the system, one associates through this to d*H* a vector field whose flow describes precisely the evolution of the mechanical system with energy *H* (e.g. Arnol'd 1995). For more complex systems, still finite-dimensional, one uses the more general notion of a Poisson manifold (Vaisman 1994). For the infinite-dimensional case, which will not be discussed here, see Dickey (2003).

### (a) Poisson manifolds

Let *M* be a differentiable manifold and let {.,.} be a Lie algebra structure on (*M*), the algebra of smooth functions on *M*. One says that {.,.} is a Poisson bracket on *M* when it is a derivation in each of its arguments (biderivation), i.e. for any *H*∈(*M*), it satisfies the Leibniz ruleThe derivation (vector field) _{H}, defined by _{H}≔{.,*H*} is then called the Hamiltonian vector field associated to *H*. Functions *H* for which _{H}=0 are called Casimirs.

It is clear that the notion of a Poisson manifold makes sense also for complex manifolds (taking for (*M*) the algebra of holomorphic functions), for complex affine varieties, that are possibly singular (taking for (*M*) the algebra of regular functions), and so on. This will be relevant later in this text.

It follows easily from the definition that the Hamiltonian vector fields form a Lie subalgebra Ham(*M*) of the Lie algebra (*M*) of all vector fields on *M* and that the Casimirs form a subalgebra of (*M*), denoted Cas(*M*). Considering the bracket on (*R*^{2n}), defined by the formula(2.1)where *q*_{1}, *p*_{1}, …, *q*_{n}, *p*_{n} are coordinates on *R*^{2n}, we recover Hamilton's equations of motionin their most symmetric formwhere is the Hamiltonian, (the sum of the kinetic and potential energy expressed in terms of position *q* and momentum *p*). It is clear that, in general, the constants of motion, i.e. the functions which are constant on the integral curves of _{H}, are the functions *F*for which {*F*, *H*}=0. Finding such functions is important for integrating the equations of motion, as we will see.

As suggested earlier, a symplectic manifold (*M*, *ω*) is a Poisson manifold. The Poisson bracket is defined for *F*, *G*∈(*M*) by , where the vector field _{H} is defined for symplectic manifolds by where . It follows easily that , so that this definition of _{H} is actually consistent with the one given earlier. The Jacobi identity for {.,.} is a consequence of the fact that *ω* is closed.

The dual ^{*} of a (finite-dimensional) Lie algebra has a natural Poisson structure that derives from the Lie bracket [.,.]. For smooth functions *F*, *G* on ^{*} the Poisson bracket is defined at *ξ*∈^{*} by(2.2)In this formula, d*F*(*ξ*) and d*G*(*ξ*), which belong to (^{*})^{*}, are interpreted as elements of when computing the bracket [.,.]. The Jacobi identity for {.,.} follows from the Jacobi identity for [.,.]. This Poisson structure is called the Lie–Poisson structure on ^{*}. When is semi-simple one can use the Killing form to transfer it to get a Poisson structure on .

The Hamiltonian vector fields define a generalized distribution, which is integrable, i.e. it admits a (unique) integral manifold through each point. These integrable manifolds carry a Poisson structure which is symplectic. In the case of the dual of a Lie algebra, for example, these leaves are precisely the coadjoint orbits.

For *m*∈*M*, the (even) dimension of is denoted by and is called the rank of {.,.} at *m*. The rank of {.,.}, denoted by , is the maximum of all ranks with *m*∈*M*. For *s*∈** N**, we denote by

*M*

_{(s)}the subsetA Poisson manifold of constant rank is called a regular Poisson manifold.

It can be shown that each of the subsets *M*_{(s)} is open and invariant for the flows of all Hamiltonian vector fields _{H}.

The Poisson structure that corresponds to a symplectic structure is regular. Indeed, its rank at any point is equal to the dimension of the symplectic manifold. Moreover, the latter property characterizes symplectic manifolds. For the Lie–Poisson structure on ^{*}, the rank at a point *m* is the dimension of the coadjoint orbit that passes through *m*, so ^{*} is only a regular Poisson manifold when is Abelian.

If the rank is constant in the neighbourhood of a point *m*∈*M*, say it is equal to 2*r*, then there exist local coordinates (*q*_{1}, …, *q*_{r}, *p*_{1}, …, *p*_{r}, *z*_{1}, …, *z*_{s}) around *m*, such that, on this neighbourhood, the Poisson bracket takes the following canonical form,where 1≤*i*,*j*≤*r* and 1≤*k*<*l*≤*s*. Such coordinates are called Darboux coordinates. For a symplectic manifold, the existence of Darboux coordinates implies that a symplectic manifold has no local invariants. Clearly, this is not true for general Poisson manifolds.

### (b) Liouville integrability

We have seen that constants of motion are functions that Poisson-commute with the Hamiltonian (functions that Poisson-commute are often said to be in involution). Liouville's fundamental observation was that if one is given *n* independent functions on *R*^{2n}, which are (pairwise) in involution with respect to the Poisson bracket (2.1), then the equations of motion can be integrated by quadratures. In the general context of Poisson manifolds, this is stated in the following theorem.

*Let* (*M*, {.,.}) *be a Poisson manifold of rank* 2*r* *and suppose that F*_{1}, …, *F*_{s} *are s*=dim *M*−*r* *functions in involution*. *Let m*∈*M*_{(r)} *and suppose that we are given a system of coordinates around m*. *If the differentials* d*F*_{1}, …, d*F*_{s} *are independent at m then the integral curve*, *starting at m*, *of each of the Hamiltonian vector fields* *can be obtained locally by using only algebraic operations, the implicit function theorem and integration*.

Let us call an *s*-tuple of functions ** F**=(

*F*

_{1}, …,

*F*

_{s}) independent when the open subsetis dense in

*M*. Then the above theorem naturally leads to the following definition of integrability.

Let (*M*,{.,.}) be a Poisson manifold of rank 2*r* and let ** F**=(

*F*

_{1}, …,

*F*

_{s}) be involutive and independent, with

*s*=dim

*M*−

*r*. We say that

**is (Liouville) integrable and (**

*F**M*, {.,.},

**) is a (Liouville) integrable system. The vector fields are then called integrable vector fields and the map**

*F***is called the momentum map.**

*F*The definition also makes sense when *M* is a complex manifold (such as a smooth affine variety) equipped with a Poisson structure on its algebra of holomorphic functions.

For any fixed *α*, *β*∈** R**, consider the following two polynomial functions:where

*p*

_{1}, …,

*q*

_{2}are coordinates on

*R*^{4}. Then obviously

*H*and

*F*are independent and it follows by direct computation that they are in involution with respect to the Poisson bracket {.,.} defined in (2.1) (with

*n*=2). Letting

**≔(**

*F**H*,

*F*) it follows that is a Liouville integrable system. It describes the motion of an anisotropic harmonic oscillator in a central field. This system (actually its

*n*-dimensional generalization) was first considered by Garnier (1919) and appeared as a by-product of his study of isomonodromic deformations of differential equations. It is nowadays referred to as the Garnier system.

If is a Liouville integrable system, then the integrable vector fields define an integrable distribution *D* of rank *r* on the (non-empty) open subset , where 2*r* denotes the rank of {.,.}. For *m* in this open subset we denote by the leaf of *D* that passes through *m*. We call it the invariant manifold of ** F**, passing through

*m*. It is the connected component of that contains

*m*. The Liouville theorem gives, under some additional assumption, a precise description of the invariant manifolds.

*Let* *be a real integrable system*, *where* , *and let* , *where* 2*r* *denotes the rank of* {.,.}.

*If**is compact then it is diffeomorphic to a torus*.*If**is not compact*,*but the flow of each of the vector fields**is complete on**then**is diffeomorphic to a cylinder**R*^{r−q}×*T*^{q}(0≤*q*<*r*).

*In both cases the diffeomorphism can be chosen in such a way that the vector fields* *are mapped to linear* (*i.e. translation invariant*) *vector fields*.

In the case of a regular Poisson manifold *M* of rank 2*r*, one has that *M*_{(r)}=*M*. For generic ** c** in the image of

**, one has then that**

*F*

*F*^{−1}(

**) is a disjoint union of smooth manifolds , which support**

*c**r*everywhere independent commuting vector fields. If one of the functions

*F*

_{i}is proper, then the fibre consists of a finite number of invariant manifolds , each of which is an

*r*-dimensional torus.

If *α* and *β* are positive real constants, then the energy *H* of the Garnier system is a proper map, hence all fibres of the momentum map ** F** are compact. If we take a regular value (

*h*,

*f*) of

**for which**

*F*

*F*^{−1}(

*h*,

*f*) is not empty, then

*F*^{−1}(

*h*,

*f*) is the union of a finite number of two-dimensional tori.

A stronger statement can be made around a torus that appears as an invariant manifold. Namely, the torus has a neighbourhood which is Poisson diffeomorphic to the product of the torus and a ball of complementary dimension, where the Poisson structure on this product is such that the natural coordinates on it are Darboux coordinates. These coordinates are then called action-angle coordinates. It was shown by Duistermaat (1980) that in general *global* action-angle coordinates do not exist due to monodromy.

### (c) The Adler–Kostant–Symes theorem

The most powerful theorem for constructing integrable systems is the Adler–Kostant–Symes theorem. To be precise, it is a theorem which allows one to construct a large family of functions in involution, associated to a splitting of a Lie algebra ; in many cases it can be shown, by a separate argument, that this family of functions contains an independent subset that is large enough to insure integrability.

By a splitting of a Lie algebra , we mean a (vector space) direct sum decomposition , where _{+} and _{−} are subalgebras of ; the corresponding Lie groups will be denoted by ** G**,

*G*_{+}and

*G*_{−}. In the version that is given here, we assume that is semi-simple, so that we can identity with its dual by using the Killing form 〈.|.〉. Then the orthogonal complement of

_{−}(with respect to 〈.|.〉) is isomorphic to , hence it inherits a Poisson structure {.,.}

_{+}from the Lie–Poisson structure on .

Consider the alternative Lie bracket on , defined bywhere we have decomposed *X* and *Y* according to the above splitting of . This yields another Lie–Poisson structure on ^{*}, and hence on , denoted {.,.}_{R}. It can be shown that the restriction of {.,.}_{R} to is precisely {.,.}_{+}. For *H*∈() and *X*∈, the Killing form allows us to view d*H*(*X*)∈^{*} as an element of ; we will denote it by ∇*H*(*X*). For *ϵ*∈ and *H*∈(), we denote by *H*_{ϵ} the function defined on by .

We are now ready to state the Adler–Kostant–Symes theorem on .

*Let F and H be* ** G**-

*invariant functions on*

*and suppose that ϵ*∈

*satisfies*

*and*.

*Then*

*and*;*The Hamiltonian vector fields**and**are*,*respectively*,*given by**where*;*For X*_{0}∈*and*|*t*|*small*,*let g*_{+}(*t*)*and**g*_{−}(*t*)*denote the smooth curves in**G*_{+}*with respect to**G*_{−}*which solve the factorization problem**Then the integral curve of**which starts at X*_{0}*is given for*|*t*|*small by*

The above equations for _{H} and are called Lax equations. There is also a more intrinsic version of the Adler–Kostant–Symes theorem on ^{*} (it does not use the Killing form), but it does not lead to Lax equations.

It follows from the theorem, or by a direct computation, that the spectral invariants of *X* are preserved by the flow of _{H}, where *H* is any ** G**-invariant function on . One often speaks of isospectral flow. Lax equations become particularly interesting when the Lie algebra is a loop algebra, as we will see.

Consider the Lie algebra splitting , where (with respect to ) denotes the Lie algebra of strictly lower triangular matrices (in respect of upper triangular matrices). Letting we have that , the vector space of all strictly upper triangular matrices in . It can be shown that for any *p*≤*n* the set of *p*-band matrices in forms a Poisson subspace of . Taking *p*=1 andand we find the Hamiltonian vector field , given by the Lax equation , whereand where . The coefficients of the characteristic polynomial of *X* yield enough (independent) functions in involution to insure integrability.

## 3. Algebraic integrability

We now come to the notion of algebraic integrability. The idea is to consider complex integrable systems, whose (complex) geometry is the best possible analogue of the (real) geometry that appears in the Liouville theorem. Following the original idea of Adler and van Moerbeke, we will mainly consider the case in which the fibres of the (complex) momentum map are affine parts of complex algebraic tori, leaving the general case to the end of this section.

### (a) A.c.i. systems

We first recall that an Abelian variety is a complex torus **C**^{r}/*Λ* (*Λ* a lattice in **C**^{r}) that is algebraic, which means that it admits an embedding in some projective space **P**^{N}; when *r*>1 then most complex tori are not algebraic. The embedding can be done by using *θ* functions, which are quasi-periodic functions on **C**^{r} (periodic in *r* directions, while periodic up to an exponential factor in *r* other directions). From the geometric point of view, this means that one considers sections of a fixed (very ample) line bundle on the Abelian variety. From the analytic point of view, it means that one considers functions with a fixed pole order along some (ample) divisor on the Abelian variety. The class of integrable systems that we will consider makes it possible to construct a basis of these functions, for the families of Abelian varieties that appear as the fibres of the momentum map; no other general method for constructing such a basis is known.

Let be an integrable system, where *M* is a non-singular affine variety and * F*=(

*F*

_{1}, …,

*F*

_{s}). We say that is an algebraic completely integrable system or an a.c.i. system if for generic

*m*∈

*M*the invariant manifold is an affine part of an Abelian variety and the Hamiltonian vector fields are translation invariant, when restricted to these tori.

We restrict ourselves to a.c.i. systems which are irreducible in the sense that the generic Abelian variety of the a.c.i. system does not contain a subtorus (an Abelian variety that contains a (non-trivial) subtorus is essentially a product of this subtorus with a complementary subtorus). It turns out that many (most) of the integrable systems that were known classically, turn out to be a.c.i., when complexified. This means that the powerful tools of the theory of Abelian varieties (dating mainly from the nineteenth century) can be used to solve and study these systems.

Let us reconsider the Garnier system. *R*^{4} and the functions *H* and *F* are complexified in the obvious way. If *α*≠*β* then for generic (*h*,*f*)∈*C*^{2} the fibre *F*^{−1}(*h*,*f*)⊂*C*^{4} is an affine part of an Abelian surface of type (1,4), which means that the lattice *Λ* that defines the torus is spanned by the columns of a matrix which has the formwithpositive definite. The divisor to be added to this affine part to complete it into an Abelian surface is a Riemann surface of genus 5. The commuting vector fields _{H} and _{F} extend to linear vector fields on these Abelian surfaces, hence the Garnier system is a.c.i. Note that it follows that if *h* and *f* are generic, then the affine variety defined byis an affine part of an Abelian surface of type (1,4).

The vector fields of an a.c.i. system have good properties at infinity; after all, a linear vector field on a complex torus is the same along the divisor, which will happen to be absent in phase space, as on the rest of the torus. In fact, since every holomorphic function on a complex torus can be written as a quotient of *θ* functions, the integral curves (solutions) to any of the vector fields of an a.c.i. system can be written as a quotient of holomorphic functions. Intuitively speaking, this means that we can consider not only Taylor solutions to the differential equations that describe these vector fields but also Laurent solutions that will correspond to initial conditions at infinity (precisely: on the divisor that needs to be adjoined to the fibres of the momentum map to complete them into Abelian varieties). Moreover, these Laurent solutions must depend on dim *M*−1 free parameters, which correspond to the freedom of choice of the initial condition at infinity. This idea is due to Kowalevski, who used it to find *her* top; her idea was further developed by Adler and van Moerbeke who applied it in several other cases of interest and used the Laurent solutions to further explore the geometry of the invariant manifolds. A modern version of this idea is given in the following theorem (Adler *et al*. 2004).

*Let* *be an irreducible a.c.i. system*, *where* ** F**=(

*F*

_{1}, …,

*F*

_{s})

*is a polynomial map*,

*and let*(

*x*

_{1}, …,

*x*

_{n})

*be a system of linear coordinates on*

*C*^{n}.

*Let*

*be any one of the integrable vector fields*.

*For every*1≤

*i*≤

*n*

*such that x*

_{i}

*is not constant along the integral curves of*,

*i.e.*,

*there exists a Laurent solution*,

*depending on n*−1

*free parameters*,

*for which x*

_{i}(

*t*)

*has a pole*.

This criterion has been used to single out from a family of Hamiltonian vector fields those that are integrable, and admit a good complexification (making them the most natural members of the family). Of course in the end, one still has to show that the Hamiltonians that have been selected lead indeed to an integrable system with respect to an a.c.i. system. The most beautiful example is the following (Adler & van Moerbeke 1991).

For *l*≥1 let *e*_{0}, …, *e*_{l} be linearly dependent vectors in a Euclidean vector space , any *l* of which are linearly independent. Let us suppose that the non-zero real numbers *p*_{0}, *p*_{1}, …, *p*_{l} which satisfy have a non-zero sum, i.e. . Let *A* be the (*l*+1,*l*+1) matrix, defined byConsider the vector field on *C*^{2(l+1)} which is given bywhere *x*, *y*∈*C*^{l+1} and *x*.*y* is defined by (*x*.*y*)_{i}=*x*_{i}*y*_{i}, *i*=0, …, *l*. Theorem 3.1 can be used to show that if the vector field is one of the integrable vector fields of an irreducible a.c.i. system, then *A* is the Cartan matrix of a twisted affine Lie algebra.

An interesting feature of a.c.i. systems is that their solutions, with any initial conditions, are single-valued. The non-single-valuedness of solutions to differential equations, in general, comes from the fact that analytic continuation of solutions usually depends on the (homotopy class of) path along which the solution is analytically continued and not just on the endpoints. This leads to a second criterion for a.c.i. which was first used by Lyapunov. The proof of it is due to Haine (1984).

*Let* *be an a.c.i. system and F be an arbitrary element of* ** F**.

*All solutions to the integrable vector field*

_{F}

*are single-valued*.

*Moreover*,

*if*

*is any closed path and x*(

*t*)

*is a solution to*

_{F}

*that is holomorphic in a neighbourhood of the path, then the analytic continuation along γ of the solution to the variational equations which correspond to the solution x*(

*t*),

*is single-valued*.

This theorem seems (at least!) as powerful as the above Kowalevski–Painlevé criterion but, up to now, it has only been used in one example (also due to Haine (1984)).

Geodesic flow for a left invariant metric on a Lie group ** G** reduces to a Hamiltonian flow on its Lie algebra . The corresponding vector field is described by the following Lax equationwhere

*H*is the quadratic form on that describes the metric. It is a Hamiltonian vector field, with

*H*as Hamiltonian, the Poisson structure being the Lie–Poisson structure on ( and its dual are identified by using

*H*). Suppose now that and

*H*is diagonal in the sense that , where and

*λ*

_{ij}=

*λ*

_{ji}for 1≤

*i*<

*j*≤

*n*. Theorem 3.2 can be used to show that if all

*λ*

_{ij}are distinct, then the vector field

_{H}is a.c.i. if and only if there exists constants

*a*

_{1}, …,

*a*

_{n},

*b*

_{1}, …,

*b*

_{n}such thati.e.

*H*defines a Manakov metric on .

The above criteria for algebraic integrability are related to the non-integrability theory developed by Morales–Ruiz and Ramis (Morales Ruiz 1999), but the precise relationship still has to be worked out.

### (b) Proving a.c.i.

The problem arises to show that a given integrable system is a.c.i. In exceptional cases the proof may follow from abstract arguments, for example, the integrable system may have been constructed essentially from algebro-geometric data. Prime examples of this are the Hitchin system and its generalizations (Hitchin 1987; Donagi & Markman 1996). The construction involves beautiful geometric constructions but it is not known how to realize these systems explicitly (e.g. to write down the functions in involution, the equations of motion, …). What we wish to describe in this section is the other extreme, namely what to do when not much more is given than explicit formulae for the functions in involution and the Poisson structure (which gives the commuting vector fields). In particular it is not assumed that Lax equations for the vector field are known; they come indeed often only at the end, when the geometry of the problem has been revealed, which is done while (and by) proving a.c.i. The ideas that underlie the technique, based on the work of Adler & van Moerbeke (1989) and Adler *et al*. (2004), will be illustrated on a non-trivial example that we refer to as the KM5 lattice, which stands for *periodic 5-particle Kac*–*van Moerbeke lattice*. The latter is given by the quadratic vector fieldwhere *x*_{1}, …, *x*_{5} are coordinates on *C*^{5} and *x*_{i+5}=*x*_{i}, for *i*∈** Z**. This vector field admits the following three independent constants of motion:A vector field that commutes with is given byhaving the same constants of motion. In fact, both and are Hamiltonian with respect to the quadratic Poisson structure that is defined bywhere 1≤

*i*,

*j*≤5. Namely, we recover both vector fields by taking

*F*

_{1}and

*F*

_{2}as Hamiltonians, while

*F*

_{3}is a Casimir. Note also that both the vector fields and the constants of motion are invariant with respect to the order 5 automorphism

*σ*of

*C*^{5}, which is defined by

*σ*(

*x*

_{i})=

*x*

_{i+1}. It follows that each invariant manifold (and its compactification) admits an automorphism of order 5, also denoted by

*σ*. This has a pronounced implication for the geometry of these manifolds, as we will see.

The rank of {.,.} is 0 on the 5 two planes, which are defined by three non-consecutive *x*_{i} being 0 (i.e. the plane *x*_{1}=*x*_{2}=*x*_{4}=0 and its image, under all powers of *σ*); the rank is utmost 2 on the 10 three planes which are given by *x*_{i}=*x*_{j}=0, where 1≤*i*<*j*≤5. On the remaining dense open subset, the rank is 4. Letting it follows that is Liouville integrable.

In order to show that the KM5 lattice is a.c.i., we fix a generic ** c**∈

*C*^{3}and we show that

*F*^{−1}(

**) is an affine part of an Abelian variety, on which and restrict to linear vector fields. To do this we verify that**

*c*

*F*^{−1}(

**) satisfies the conditions of the following theorem, which is a complex analogue of the Liouville theorem.**

*c**Let* ⊂*C*^{s} *be a non-singular affine variety of dimension r which supports r holomorphic vector fields* _{1}, …, _{r} *and let* *be an embedding*. *We define* *and denote the union of all irreducible components of Δ of dimension r*−1 *by Δ*′. *Suppose the following*.

*The vector fields commute pairwise*, [_{i},_{j}]=0,*for*1≤*i*,*j*≤*r*.*At every point m*∈*the vector fields**are independent*.*The vector field**extends to a vector field**which is holomorphic on a neighbourhood of**in*.*The integral curves of**that start at points**go immediately into*.

*Then* *is an Abelian variety of dimension r and the vector fields* *extend to holomorphic* (*hence linear*) *vector fields on* . *Moreover*,

For our example, we take for the smooth manifold , where ** c** is a regular value of

**, satisfying**

*F**c*

_{3}≠0 (a generic

**∈**

*c*

*C*^{3}has these properties). Note that if is such that

*c*

_{3}≠0 then is entirely contained in , so that each invariant manifold is a connected component of a fibre of the momentum map; in fact, since is irreducible it coincides with the invariant manifold through any of its points. Thus, is a non-singular affine variety of dimension

*r*, equipped with

*r*holomorphic vector fields that satisfy conditions (i) and (ii). It therefore suffices to construct an embedding which satisfies (iii) and (iv) to conclude that the generic invariant manifold satisfies the conditions, given in the definition of a.c.i.

To do this, we use the Laurent solutions to . These are rather easily found because one can show that, in this case, each of the Laurent series has a simple pole at worst. Substitutingwe find that the leading coefficients of the series satisfiythe so-called indicial equation. It has 10 solutions, namely the pointsand their images , under repeated application of the order 5 automorphism *σ*. The other terms are then obtained recursively, but linearly, from the solutions to the indicial equation. We give the first few terms of the Laurent solution for which the leading term is given by ,(3.1)where the four free parameters have been denoted by . Using the majoration method, one shows that they are actually convergent. Using the automorphism *σ* we find in total five such Laurent solutions , depending on four free parameters. The Laurent solutions (depending on four free parameters) for (or any linear combination of and ) can be computed from it, but they will not be needed (in certain examples, such as the periodic Toda lattice, these Laurent solutions are needed to prove a.c.i.). The points lead to Laurent solutions that depend only on three free parameters; they are not used in what follows.

Since the functions are constants of motion, a direct substitution of in each of them yields a Laurent series in *t*, of which only the term in survives. Clearly, this term can be computed by using only the first terms of the series . In the present case the terms that are given, suffice to do this for each of the functions *F*_{i}. We find three algebraic relations between the free parameters and the values of the constants of motion *c*_{1}, *c*_{2}, *c*_{3}, to wit,This defines an affine algebraic curve, denoted , in *C*^{4}. Clearly, is isomorphic to the non-singular plane curveBy Riemann–Hurwitz the genus of *Γ*_{c} is 2. It is easy to see that *Γ*_{c} has three points at infinity, corresponding to *d*=0 or ∞; they will be denoted by ∞_{c}, and . In terms of a local parameter *ς*, a neighbourhood of these points is parametrized as follows:(3.2)It is clear that if we use any other principal balance , then we get an affine curve , which is also isomorphic to *Γ*_{c}. Thus, considering all five Laurent solutions that depend on four free parameters (they are obtained from the given one by using the automorphism ) we get five isomorphic curves of genus 2. Each of them is compactified into a Riemann surface by adding three points at infinity.

We now construct the embedding . The idea of the construction is based on the following fact. If KM5 is indeed an irreducible a.c.i. system, then a divisor can be added to a Zariski open subset of phase space , having the effect of compactifying all fibres , where is generic; the divisor that is added to will be denoted by and the resulting torus by . The KM5 vector fields and extend to linear (hence holomorphic) vector fields and on this partial compactification of , hence we may consider the integral curves of , starting from any irreducible component of this divisor. This gives a Laurent solution, depending on four free parameters, hence it must coincide with one of the Laurent solutions that we have computed. Let *f* be a polynomial function on . Since is transveral to the divisor the pole order of the Laurent series (which can be obtained by substituting the series in *f*) equals the pole order of along the divisor , where is by definition , viewed as a meromorphic function on . Since the third power of an ample divisor on an Abelian variety is very ample, we look for all the polynomials which have a triple pole at most when is substituted in them, and no pole at all when for is substituted in them. Precisely, we look for a maximal independent set of such functions which are independent when restricted to . One finds easily, besides the following independent functions:In the sequel we think of these functions as being restricted to , and we consider the map(3.3)Since , does not intersect any of the hyperplanes , so that is birational on its image, namelywhile is recovered from . In particular, is an embedding. The closure of (with respect to the Zariski topology, or, what is the same, with respect to the complex topology) is denoted by . Also, we denote by the union of all irreducible components of of dimension , as in theorem 3.3.

The leading terms of the above Laurent series lead to a map , given by(3.4)which is obviously an embedding (recall that on ). Similarly, the Taylor (!) series lead to four different embeddings of this curve, to wit(3.5)where we recall that and . Since we see at once that the five image curves are disjoint. However, these images are not complete, so we check if maybe their closures intersect. In order to do this, we compute the image of and under each of the above five embeddings. This is done by substituting the above local parametrizations (3.2) in the embeddings ; taking an extra term in the computation, we can also determine the tangent line to the image of to deduce from it whether these closures are non-singular and how they intersect (table 1).

In this table, the points are the following points in :Denoting by , we find that contains the points and and that each intersects its neighbour in two different points and , while being tangent to the divisors and . The resulting divisor is depicted in figure 1. We need to show that extends to a holomorphic vector field on a neighbourhood of in (condition (iii) of theorem 3.3). Note that we do not know yet whether coincides with , so we do not really have control over . In fact, we will be able to show that extends to a holomorphic vector field on *all* of , which implies in particular holomorphicity in a neighbourhood of , whatever the latter may be. The construction of the extension is based on the following theorem, which says that in appropriate projective coordinates, any holomorphic vector field on an Abelian variety is a quadratic vector field, hence it is globally defined and holomorphic (on all of ).

*Let* *be an ample line bundle on an irreducible Abelian variety* *and let* *be a holomorphic vector field on* . *Denoting by* *the Kodaira embedding that corresponds to* , *there exists a quadratic vector field* *on* , *such that* .

Thus, even if we do not know yet that our example is a.c.i., we look for quadratic differential equations on , whose restriction to describes . This can be done in an algorithmic way, because and all invariants are homogeneous. In the chart of , we arrive at the following1 result.In the same way one easily finds the differential equations in another chart. In view of Hartog's theorem, we may conclude that the vector field extends to a holomorphic vector field on , which we denote by . Thus, we have verified that satisfies condition (iii) in the Complex Liouville theorem.

The final thing to be shown is that the integral curves of that start at points go immediately into . We repeat that we do not know if . Therefore, there are three types of points that we need to consider.

Points in the image , where .

Points in , where .

Points in .

For (i) one uses that the embedding functions are polynomials in the phase variables and the fact that the Laurent series are convergent to conclude that, for small , all series are finite, hence the flow starting from the points as in (i) does not belong to the hyperplane . Similarly all quotients are finite, so we do not flow into any of the other . For (ii), we do the check for . Since the only non-zero entry corresponds to , we substitute the parametrization (3.2) of a neighbourhood of into . Since we have that(3.6)which yields, in the limit , the series , which is different from zero. This means that starting from , we do not flow into the divisor . In order to show that we flow into the affine , we show that and have also a non-zero limit, as . For the first one, this is trivial sinceas follows from (3.1). For the second one, the computation is longer, since the first terms of the series vanish when taking the limit; besides this fact, which only makes the computation longer, the calculation is trivial and we only state the final formula,This shows that we flow into the affine, starting from .

Finally, we turn to the points (iii). We show that there are no such points. Since is irreducible and is regular, the divisor is connected. Therefore, if contains irreducible components that are different from the , then at least one of the former ones must intersect one of the latter ones. Moreover, this must happen at points where the series break down, i.e. at the points (ii). We can however compute the degree of at these points. We do this for the point . Letting we have that is a defining function for the divisor around and since is transveral to at , we can read off from (3.6) that the degree of . Since passes through and since this divisor is taken with multiplicity 3, the divisor that we have found already accounts for the degree three and there cannot be another divisor, passing through . For the points one proceeds in the same way, thereby concluding the proof that KM5 is a.c.i.

Since the torus contains a curve of genus 2, it is the Jacobian of this curve and this curve is a translation of the *θ* divisor. Note that three times the *θ* divisor induces on the Jacobian a polarization of type , which leads to an embedding of the Jacobian in ; the above embedding is a concrete realization of this. For an alternative characterization of these tori in terms of hyperelliptic Prym varieties, and for an alternative proof of algebraic integrability of KM5 (in fact of all KM), see Fernandes & Vanhaecke (2001). For more elaborate examples and a justification of the facts that are used in the above proof, see Adler *et al*. (2004).

### (c) Lax equations with a parameter

The integrable vector fields of most a.c.i. systems can be written down in the form of a Lax equation with a parameter, defined as follows.

Let *M* be a finite-dimensional affine subspace of the Lie algebra , let be a Poisson structure on *M* and let . If the Hamiltonian vector field on *M* can be written in the form(3.7)where is a function on *M* with values in , then (3.7) is called a Lax equation with parameter for .

As in the case of Lax equations (without a parameter), one has that the functions which are defined by the coefficients of the characteristic polynomial of are constants of motion of (3.7). Therefore, if we associate to each , the plane algebraic curvewe have that is preserved by the flow of (3.7). Similarly, for each the variety of matricesis preserved by the flow of (3.7). Each of the curves is called a spectral curve and each of the varieties is an isospectral variety; when is smooth, we also use the name isospectral manifold. Let us suppose that for generic , the affine curve is non-singular and that for generic , the eigenspace of with eigenvalue is one-dimensional. Let (possibly after some relabelling)be the eigenvector which is normalized at . Then each is a rational function in , whose coefficients are functions on *M*. When flows according to (3.7), the normalized eigenvector also evolves, satisfying the autonomous equationwhere is also a rational function whose coefficients are functions on *M*.

The isospectral manifolds and isospectral curves are linked by a map that we introduce next. For simplicity we will assume that is chosen such that is non-singular and is connected. Since is non-singular, it can be completed into a compact Riemann surface by adding a few points, which will be denoted by . For and an open subset of denote by the divisor of zeros and poles of , restricted to . Since depends in a holomorphic way on *t* (for small), we have that for a generic , with corresponding normalized eigenvector , the minimal effective divisor on which satisfieshas a degree *d* which is independent of . This leads, by continuity, to an effective divisor of degree *d* in for any . The resulting mapis called the divisor map. When evolves according to (3.7), the image of under evolves on ; we will denote by . Choose a basis of , where *g* is the genus of and let . The map , defined by(3.8)is called the linearizing map, starting at (with respect to ). The following theorem gives a criterion to check whether this map linearizes the isospectral flow.

*Suppose that* *is chosen such that* *is smooth*, *and let* *be an element of* . *The linearizing map* *linearizes the isospectral flow defined by* (3.7) *if and only if there exists a meromorphic function* on *with* , *and such that for all* (3.9)

For a list of examples that satisfy the linearization criterion, see Adler & van Moerbeke (1980, p. 302, theorem 4.3). Our example 2.6 is easily adapted to write the KM5 vector field as a Lax equation (with parameter) , by takingIn this case all Laurent tails are constant at infinity, so KM5 satisfies the linearization criterion.

### (d) Other notions of algebraic integrability

For certain examples of interest, the generic invariant manifold is not an affine part of an Abelian variety, but still the flow of the integrable vector fields induces on them the action of a local group and the group law is algebraic. In order to cover these cases, we propose here the following definition that generalizes definition 3.1.

Let be a complex integrable system, where *M* is a (non-singular) affine variety and . We say that is a generalized a.c.i. system if for generic , the integrable vector fields define the local action of an algebraic group on .

Note that the algebraic group is necessarily commutative because the vector fields commute pairwise. The case of an a.c.i. system is a particular example: in this case the algebraic group is a compact algebraic group, i.e. an Abelian variety. We give two examples where the group is not compact.

Let be a compact-oriented topological surface of genus with fundamental group and let be a reductive algebraic group. Then is an affine variety on which acts by conjugation, more precisely if and then is the homomorphism defined byfor . It turns out (Goldman 1986) that the quotient(which is an affine variety since is reductive) has a natural Poisson structure which can very explicitly be described for the classical groups. For simplicity let us consider the case in the standard representation. For a curve the functionis a well-defined regular function on *M* and these functions generate . It was shown by Goldman that on such functions a Poisson bracket of maximal rank is given by(3.10)The sum runs over the intersection points of *C* and (one may suppose that the curves intersect transversally) and is a sign which is determined by the way the (oriented) curves *C* and intersect at *p*, upon using the orientation of . Finally, is the curve on , based at *p*, which is obtained by first following *C* and then following .

Let us consider the case . Since has a system of generators, which are bound by one relation, has dimension , hence *M* has dimension . Since the rank of the Poisson structure is maximal, we need to find independent functions on *M* that are in involution. Now can be decomposed (in an infinite number of different ways) into so-called trinions; a trinion, also called a pair of pants, is just a three-holed sphere and such a decomposition will consist of trinions. Each trinion being bounded by three curves (which are identified two by two) one gets curves on and what is important here is that they are non-intersecting. Calling these curves , we find from Goldman's formula (3.10) that the functions are in involution. They can be shown to be independent, hence they lead to an integrable system on *M*. The fibres of the momentum map are in this case toric varieties and (for a generic point in *M*) the non-compact algebraic group is .

In the following example, the algebraic group is a mixture of the two types of algebraic groups considered up to now, namely it is an extension of an Abelian variety with several copies of .

The Lagrange top is by definition a rigid body with a fixed point (‘spinning top’) which is symmetric with respect to an axis passing through the centre of gravity and the fixed point. It admits a complexification, which is the integrable system, defined by the Hamiltonianwhere are coordinates on . The Poisson structure, which comes from the Lie–Poisson structure of , the Lie algebra of the group of Euclidean motions in three-space, is given bywhere and is the totally anti-symmetric tensor for which . The constants are the moments of inertia and they satisfy ; the constant *m* is given by . It follows that the Hamiltonian vector field is given, up to a constant, byOne shows by direct computation that, besides the Hamiltonian, one has the following three constants of motion, making the Lagrange top integrable,

It is shown (Gavrilov & Zhivkov 1998) that, in this form, the Lagrange top is a generalized a.c.i. system. The following characterization is given: the generic level of the momentum map is isomorphic to an affine part of the generalized Jacobian of an elliptic curve, which two points (at infinity) identified. Thus, the algebraic group in this case is a (non-trivial) *C*^{*}-extension of an elliptic curve (which is a one-dimensional Abelian variety).

Further examples of a.c.i. and generalized a.c.i. systems are the periodic and non-periodic Toda lattices, the periodic and non-periodic Kac–van Moerbeke lattices, the Mumford system and some of its generalizations, the Hitchin system and its generalizations, the classical integrable tops (with the exception of the Goryachev–Chaplygin top), three families of geodesic flows on ** SO**(4), and many others (Adler

*et al*. 2004 and references therein). For a large family of polynomial integrable systems, associated to an arbitrary algebraic curve, but which are not (generalized) a.c.i., see Vanhaecke (2001).

## Footnotes

One contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.

↵The result is not unique because one can always add to the right-hand side any quadratic polynomial that vanishes on the Abelian variety; we have used this fact to keep the formulas as simple as possible.

- © 2007 The Royal Society