A Hamiltonian system on a Poisson manifold $M$ is called integrable if it possesses sufficiently many commuting first integrals $f_1, \ldots f_s$ which are functionally independent on $M$ almost everywhere. We study the structure of the singular set $K$ where the differentials $df_1, \ldots, df_s$ become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers.

We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.

A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on $e(3)$ parametrized by polynomial a with the above Lax matrices are constructed. Five cases from the family are selected by the condition of preserving the standard measure. Three of them are Hamiltonian. It is proved that two other cases are not Hamiltonian in the standard Poisson structure on $e(3)$. Integrability of all five cases is proven. Integration procedures are performed in all five cases. Separation of variables in Sklyanin sense is also given. A connection with Hess-Appel’rot system is established. A sort of separation of variables is suggested for the Hess-Appel’rot system.

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

We revise the solution to the problem of Hamiltonization of the $n$-dimensional Veselova non-holonomic system studied previously in [1]. Namely, we give a short and direct proof of the hamiltonization theorem and also show the trajectorial equivalence of the problem with the geodesic flow on the ellipsoid.

We consider the class of ellipsoidal gas clouds expanding into a vacuum [1, 2] which has been shown to be a Liouville integrable Hamiltonian system [3]. This system presents several interesting features, such as the Painlevé property [4, 5], the existence of Bäcklund transformations and the separability of variables, all shown to be present in at least several sub-cases.
A remarkable result that emerged from the study of the cases of rotation around a fixed principal axis, was that the Liouville torus, which is the locus of trajectories of the representative point of the cloud when all the constants of motion are fixed, could be assimilated with a quartic surface presenting 16 conic point singularities. The geometry of such surfaces is entirely determined by the datum of a 6th degree polynomial in one variable, and the consideration of the corresponding natural coordinate system then led to the separation of variables for these cases [6]. Further, the equation of the surface takes the form of a $4 \times 4$ determinant, which constitutes a generalization of Stieltjes $4 \times 4$ determinant formulation of the addition formula for elliptic functions; and the corresponding matrix also defines the system of the equations of motion; so that it can be said that the differential system is completely determined by the surface’s geometry.
Forsaking now the assumption of a fixed rotation axis, in cases where the energy constant takes its minimum value compatible with the other constants of motion, we found that the Liouville torus was still reducible to the form of a quartic surface, presenting 15 conic points only instead of 16 (16 conic points were indeed present originally, but one of them had to disappear in the process of reducing the surface to the 4th degree). The geometry of these surfaces is entirely determined by the datum of a plane unicursal quartic (which is the transformed version of the missing conic point). The system can be reduced to the form of a differential equation of second degree, the coefficients of which are polynomials of degree 7, which are determined by the surface’s geometry, except for their quadratic dependence on a single free parameter, $z$. Defining $u$ the (time-like) independent variable, and $\Phi$ the integration constant (which are functions defined on the Liouville torus), it is found that $\Phi$ depends linearly on the parameter: $\Phi = \Phi(z)$ and then u may be taken to coincide with $\Phi(z')$, for any value of $z'$ distinct from $z$. Solving the system for one particular value of $z$ therefore also solves it for all other values of the parameter. It appears that the geometry alone does not specify in this case any particular value of $z$, but then any two values lead to differential systems which (although their solutions differ) turn out to be equivalent. It may also be worth pointing out that changing $z$ may be viewed as exchanging the roles of $u$ and $\Phi$.
Finally, in degenerate cases the Liouville torus presents a double line of self-intersection, and the separation of variables can be achieved. Sections by planes through the double line are conic sections, which may be labeled by a parameter w, say. Denoting α the eccentric anomaly on the conic, the differential system in fact takes a remarkably simple form: $da/dw = f(w)$, and involves an elliptic inteqgral.

We apply a non-linear matrix transformation of Lie–Bäcklund type on a seed soliton configuration in order to obtain a new solitonic solution in the framework of the 5D low-energy effective field theory of the bosonic string. The seed solution represents a stationary axisymmetric two-soliton configuration previously constructed through the inverse scattering method and consists of a massless gravitational field coupled to a non-trivial chargeless dilaton and to an axion field endowed with charge. We apply a fully parameterized non-linear matrix transformation of Ehlers type on this massless solution and get a massive rotating axisymmetric gravitational soliton coupled to charged axion and dilaton fields. We discuss on some physical properties of both the initial and the generated solitons and fully clarify the physical effect of the non-linear normalized Ehlers transformation on the seed solution.

The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.

We discuss some special classes of canonical transformations of the time variable, which relate different integrable systems. Such dual systems have different integrals of motion, Lax equations, separated variables and bi-hamiltonian structures. As an example the two-dimensional periodic Toda lattices associated with the classical root systems and the dual natural systems on the palne are considered in detail.

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems $H = \frac{1}{2} p^2 + V(q)$ is explained in the first part of this review. It has a form of an effective criterion that for any given potential $V(q)$ tells whether there exist suitable separation coordinates $x(q)$ and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials $V(q)$ all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrodinger equation of quantum mechanics.
Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $\ddot q=M(q)$ admitting $n$ quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $\ddot q=−∇V(q)$. The cofactor pair Newton equations admit a Hamilton–Poisson structure in an extended $2n + 1$ dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.

Four new integrable classes of mechanical systems on Riemannian 2D manifolds admitting a complementary quadratic invariant are introduced. Those systems have quite rich structure. They involve 11–12 arbitrary parameters that determine the metric of the configuration space and forces with scalar and vector potentials. Interpretations of special versions of them are pointed out as problems of motions of rigid body in a liquid or under action of potential and gyroscopic forces and as motions of a particle on the plane, sphere, ellipsoid, pseudo-sphere and other surfaces.

In this issue we bring to the reader’s attention a translation of Levi-Civita’s work "Sulle trasformazioni delle equazioni dinamiche". This paper, written by Levi-Civita at the onset of his career, is remarkable in many respects. Both the main result and the method developed in the paper brought the author in line with the greatest mathematicians of his day and seriously influenced the further progress of geometry and the theory of integrable systems. Speaking modern language the main result of his paper is the deduction of the general geodesic equivalence equation in invariant form and local classification of geodesically equivalent Riemannian metrics in the case of arbitrary dimension, i.e., metrics having the same geodesics considered as unparameterized curves (this classification problem was formulated by Beltrami in 1865). Levi-Civita’s work produced a great impact on further development of the theory of geodesically equivalent metrics and geodesic mappings, and still remains one of the most important tools in this area of differential geometry.
In this paper the author uses a new method based on the concept of Riemannian connection, which later has been also referred to as the Levi-Civita connection. This paper is truly a pioneering work in the sense that the real power of covariant differentiation techniques in solving a concrete and highly nontrivial problem from the theory of dynamical systems was demonstrated. The author skillfully operates and weaves together many of the most advanced (for that times) algebraic, geometric and analytic methods. Moreover, an attentive reader can also notice several forerunning ideas of the method of moving frames, which was developed a few decades later by E. Cartan.
We hope that the reader will appreciate the style of exposition as well. This work, focused on the essence of the problem and free of manipulation with abstract mathematical terms, is a good example of a classical text of the late 19th century. Owing to this, the paper is easy to read and understand in spite of some different notation and terminology.
The Editorial Board is very grateful to Professor Sergio Benenti for the translation of the original Italian text and valuable comments (see marginal notes at the end of the text, p. 612).