The article continues the investigations of bifurcation diagrams of the Kowalevski top in two constant fields started by the first author in [10], [1]. We analyze the admissible values of two almost everywhere independent integrals arising on the invariant submanifold that generalizes the so-called 4th Appelrot class of the Kowalevski top in the field of gravity.

For construction and classification of the natural integrable systems we propose to use a criterion of separability in Darboux–Nijenhuis coordinates, which can be tested without an a priori explicit knowledge of these coordinates. As an example we apply this method for the search of integrable systems on the plane.

This note constructs a compact, real-analytic, riemannian 4-manifold $(\sum, \mathscr{g})$ with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) $\sum$ is diffeomorphic to ${\bf T}^2 \times {\bf S}^2$; and (3) the limit set of the geodesic flow on the universal cover is dense. This shows there are obstructions to real-analytic integrability beyond the topology of the configuration space.

The article is devoted to examining algebraic closure relationships which are used in the Theory of Semiempirical Models of Turbulence. As an example, the dynamics of a far plan turbulent wake is investigated and the so-called locally equilibrium approximation of second-order moments (tangential Reynolds stresses) is considered. Applicability of this algebraic approximation for tangential Reynolds stresses is analyzed by the method of differential constraints in the context of investigation of compatibility of the original mathematical model (the classical ($e$, $\varepsilon$, $ < u' v' >$) — model of turbulence) with an added differential constraint (i.e. with an algebraic closure relationship for tangential Reynolds stresses). We show that the compatibility condition obtained coincide with the condition that a Hamiltonian vector field generated by the velocity field of the turbulent flow under consideration admits a symplectic symmetry of the canonical transformations.

Studies of the properties of vortex motions in a stably stratified and fast rotating fluid that can be described by the equation for the evolution of a potential vortex in the quasi-geostrophic approximation are reviewed. Special attention is paid to the vortices with zero total intensity (the so-called hetons). The problems considered include self-motion of discrete hetons, the stability of a solitary distributed heton, and the interaction between two finite-core hetons. New solutions to the problems of three or more discrete vortices with a heton structure are proposed. The existence of chaotic regimes is revealed. The range of applications of the heton theory and the prospects for its future application, particularly in respect, to the analysis of the dynamic stage in the development of deep ocean convection, are discussed.