In this paper, I present an overview of the active area of algebraic completely integrable systems in the sense of Adler and van Moerbeke. These are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). We make, via the Kowalewski–Painlevé analysis, a study of the level manifolds of the systems. These manifolds are described explicitly as being affine part of abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler–Van Moerbeke method’s which will be used is devoted to illustrate how to decide about the algebraic completely integrable Hamiltonian systems and it is primarily analytical but heavily inspired by algebraic geometrical methods. I will discuss some interesting and well known examples of algebraic completely integrable systems: a five-dimensional system, the Hénon–Heiles system, the Kowalewski rigid body motion and the geodesic flow on the group $SO(n)$ for a left invariant metric.

We give a birational morphism between two types of genus 2 Jacobians in $\mathbb{P}^{15}$. One of them is related to an Algebraic Completely Integrable System: the Geodesic Flow on $SO(4)$, metric II (so termed after Adler and van Moerbeke). The other Jacobian is related to a linear system in $|4\Theta|$ with 12 base points coming from a Göpel tetrad of 4 translates of the $\Theta$ divisor. A correspondence is given on the base spaces so that the Poisson structure of the $SO(4)$ system can be pulled back to the family of Göpel Jacobians.

In this paper we review a recently introduced method for solving the Hamilton–Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.

We give an overview of the integrability of the Hirota–Kimura discretizationmethod applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota–Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.

This paper is a further contribution to the study of exact solutions to KP, KdV, sine-Gordon, 1D Toda and nonlinear Schrödinger equations. We will be uniquely concerned with algebro-geometric solutions, doubly periodic in one variable. According to (so-called) Its-Matveev’s formulae, the Jacobians of the corresponding spectral curves must contain an elliptic curve X, satisfying suitable geometric properties. It turns out that the latter curves are in fact contained in a particular algebraic surface $S\perp$, projecting onto a rational surface $\tilde{S}$. Moreover, all spectral curves project onto a rational curve inside $\tilde{S}$. We are thus led to study all rational curves of $\tilde{S}$, having suitable numerical equivalence classes. At last we obtain $d$ - 1-dimensional of spectral curves, of arbitrary high genus, giving rise to KdV solutions doubly periodic with respect to the d-th KdV flow $(d \geqslant 1)$. Analogous results are presented, without proof, for the 1D Toda, NL Schrödinger an sine-Gordon equation.

We examine a class of Lotka–Volterra equations in three dimensions which satisfy the Kowalevski–Painlevé property. We restrict our attention to Lotka–Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painlevé analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions. We also show that the conditions are sufficient.

In this paper, we consider the Toda lattice associated to the twisted affine Lie algebra $\mathfrak{d}_3^{(2)}$. We show that the generic fiber of the momentum map of this system is an affine part of an Abelian surface and that the flows of integrable vector fields are linear on this surface, so that the system is algebraic completely integrable. We also give a detailed geometric description of these Abelian surfaces and of the divisor at infinity. As an application, we show that the lattice is related to the Mumford system and we construct an explicit morphism between these systems, leading to a new Poisson structure for the Mumford system. Finally, we give a new Lax equation with spectral parameter for this Toda lattice and we construct an explicit linearization of the system.

We investigate geometric properties of the (Sato–Segal–Wilson) Grassmannian and its submanifolds, with special attention to the orbits of the KP flows. We use a coherentstates model, by which Spera and Wurzbacher gave equations for the image of a product of Grassmannians using the Powers–Størmer purification procedure. We extend to this product Sato’s idea of turning equations that define the projective embedding of a homogeneous space into a hierarchy of partial differential equations. We recover the BKP equations from the classical Segre embedding by specializing the equations to finite-dimensional submanifolds.
We revisit the calculation of Calabi’s diastasis function given by Spera and Valli again in the context of $C^*$-algebras, using the $\tau$-function to give an expression of the diastasis on the infinitedimensional Grassmannian; this expression can be applied to the image of the Krichever map to give a proof of Weil’s reciprocity based on the fact that the distance of two points on the Grassmannian is symmetric. Another application is the fact that each (isometric) automorphism of the Grassmannian is induced by a projective transformation in the Plücker embedding.

The paper revises the explicit integration of the classical Steklov–Lyapunov systems via separation of variables, which had been first made by F. Kötter in 1900, but was not well understood until recently. We give a geometric interpretation of the separating variables and then, applying the Weierstrass hyperelliptic root functions, obtain explicit theta-function solution to the problem. We also analyze the structure of poles of the solution on the Jacobian on the corresponding hyperelliptic curve. This enables us to obtain a solution for an alternative set of phase variables of the systems that has a specific compact form.
In conclusion we discuss the problem of integration of the Rubanovsky gyroscopic generalizations of the above systems.

New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.