The paper develops an approach to the proof of the «zeroth» law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the average energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.

We classify quadratic Poisson structures on $so^*(4)$ and $e^*(3)$, which have the same foliations by symplectic leaves as canonical Lie-Poisson tensors. The separated variables for some of the corresponding bi-integrable systems are constructed

Problem of sound propagation in a range-dependent underwater sound channel is studied in the scope of the problem of the ray-wave correspondence. Small-scale vertical oscillations of a sound channel inhomogeneity act on near-axial rays in a resonant way. Scattering of rays on resonance leads to forming of a wide chaotic layer with fast mixing in the underlying phase space. The Husimi distribution function is used for examining of dynamics of wavepackets belonging to the chaotic layer. At high frequencies of a signal, a wavepacket diverges rapidly with range. Decreasing of frequency leads to suppressing of resonance induced by vertical oscillations of an inhomogeneity, wavepacket stops diverging and its width in the action space starts to oscillate irregularly. At the frequency of 50 Hz these oscillations are regular, that indicates suppression of chaotic diffusion.

The conditions which are necessary for chiral-type systems to admit the Lax representation with compact Lie algebras are obtained in this paper. We also establish new integrable systems which are similar to WZNW (Wess-Zumino-Novikov-Witten) systems and non-abelian affine Toda models. One of such a system is a new integrable extension of the well known sin-Gordon equation.

Exact solutions of the Landau-Lifshits equation for a uniaxially anisotropic nondissipative ferromagnet are derived. The solutions belong to the class of monoazimuthal separable ones. The new solution describing oscillations of a 360-degrees domain wall is obtained. The results are true for biaxial ferromagnet as well.

The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of n antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases n=2,3 are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.