We show that almost all billiard trajectories return parallel to themselves for rank $1$, ergodic polygons. Applications are given to the existence of periodic trajectories.

We present a detailed study of the properties of two q-discrete Painlevé IV equations: singularity structure, bilinear forms, auto-Bäcklund/Schlesinger transformations, rational solutions as well as special solutions obtained through second- or first-order linear equations.

The theoretical study of steady vortex motion of homogeneous ideal heavy fluid with a free surface by methods of differential geometry is presented. The main idea of methods is based on suggestion that a velocity field is formed by geodesic flows at some surfaces. For steady flow integral flow lines are geodesics on the second order surfaces being parameterized and located in the space occupied by the fluid. In this case both Euler and continuity equations are transformed into equations for inner geometry parameters. Conditions on external parameters are derived from boundary conditions of the problem. The investigation of properties of generalized Rankine vortex that is vertical vortex flow contacting with a free surface is done. In supplement to the classical Rankine vortex these vortices are characterized by all finite specific integral invariants. The constructed set of explicit solutions depend on a unique parameter, which can be defined experimentally through measurements of depth and shape of a near surface hole produced by the vortex.

A collisionless continuous medium in Euclidean space is discussed, i.e. a continuum of free particles moving inertially, without interacting with each other. It is shown that the distribution density of such medium is weakly converging to zero as time increases indefinitely. In the case of Maxwell's velocity distribution of particles, this density satisfies the well-known diffusion equation, the diffusion coefficient increasing linearly with time.

We consider a class of third order homogenous differential equation whose coefficients are analytic functions of the complex variable. We find necessary and sufficient conditions about the coefficients in order that the only movable singularities of the solutions may be poles.

Planar librations of a satellite, with its mass center moving in an elliptic orbit, are under consideration. Besides the gravitational force, the satellite is acted on by the Solar light pressure.
Rotation of a dynamically symmetrical satellite is taken as the unperturbed system. A direct application of the KAM-theorem is impossible because of nonanalyticity of the Hamiltonian. Using the reduction of the perturbed Hamiltonian system to a sequence of symplectic maps, and with the help of Moser's theorem on invariant curve, an existence of invariant tori and the fact that the action variables remain close to their initial values are proven.
The vicinity of the limit case (the orbit eccentricity is equal to or approximately equal to $1$) is also studied. In this case, the order of perturbation is supposed to be fixed. It turns out that in this case the action variables also preserve their values over asymptotically large time intervals.

The problem on inclusion of an invariant manifold of a dynamical system into a set of integral manifolds is considered. It is shown that this inclusion is always possible unless the invariant manifold is a singular one (consisting of singular points of the system) of codimension one. It enables us to study invariant manifolds using the equation for integrals instead of the Levi-Civita equations containing uncertain factors. The defining role of the singular manifolds in shaping a phase portrait of a dynamical system is established and the following from this integrals properties are obtained. The results are applied to the analysis of solutions of the Euler–Poisson equations. A new treatment of the fourth integrals in Euler's, Lagrange's, Kovalevskaya's cases is proposed. It is proved that under the Hess conditions there is a fourth integral of which special cases are Euler's and Lagrange's integrals as well as the Hess and the Dokshevich solutions.