Abstract. The problem of the geodesic curves on a closed two-dimensional surface and some of its generalizations related with the addition of gyroscopic forces are considered. The authors study one-parameter groups of symmetries in the four-dimensional phase space that are generated by vector fields commuting with the original Hamiltonian vector field. If the genus of the surface is greater than one, then there are no nontrivial symmetries. For a surface of genus one (a two-dimensional torus) it is established that if there is an additional integral polynomial in the velocities, even or odd with respect to each component of the velocity, then there is a polynomial integral of degree one or two. For a surface of genus zero examples of nontrivial integrals of degree three and four are given. Fields of symmetries of first and second degree are studied. The presence of such symmetries is related to the existence of ignorable cyclic coordinates and separated variables. The influence of gyroscopic forces on the existence of fields of symmetries with polynomial components is studied.

Abstract. The geodesic curves of a Riemannian metric on a surface are described by a Hamiltonian system with two degrees of freedom whose Hamiltonian is quadratic in the momenta. Because of the homogeneity, every integral of the geodesic problem is a function of integrals that are polynomial in the momenta. The geodesic flow on a surface of genus greater than one does not admit an additional nonconstant integral at all, but on the other hand there are numerous examples of metrics on a torus whose geodesic flows are completely integrable: there are polynomial integrals of degree one or two that are independent of the Hamiltonian. It appears that the degree of an additional ''irreducible'' polynomial integral of a geodesic flow on a torus cannot exceed two. In the present paper this conjecture is proved for metrics which can arbitrarily closely approximate any metric on a two-dimensional torus.

Abstract. The problem of motion of a point in an n-dimensional sphere under the action of a force with meromorphic components is considered. The relation between the number of the integrals polynomial with respect to the velocities and the number of poles of the force moment is established. The conditions under which the problem solutions branch in the plane of complex time are found.

Abstract. The problem of geodesic lines on a two-dimensional torus is considered. One-parameter symmetry groups in the four-dimensional phase space that are generated by vector fields commuting with the initial Hamiltonian vector field are studied. As proved by Kozlov and Bolotin, a geodesic flow on a two-dimensional torus admitting a non-trivial infinitesimal symmetry of degree n has a many-valued integral that is a polynomial of degree at most n in the momentum variables. Kozlov and the present author proved earlier that first- and second-order infinitesimal symmetries are related to hidden cyclic coordinates and separated variables. In the present paper the structure of polynomial infinitesimal symmetries of degree at most four is described under the assumption that these symmetry fields are non-Hamiltonian.

Abstract. We show in this paper that if a geodesic flow on a two-dimensional torus admits a non-Hamiltonian field of third degree symmetries in momenta, then the conformal multiplier of a metric on the two-dimensional torus satisfies a certain nontrivial relation.

Abstract. We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an ''irreducible'' integral of the geodesic flow on the torus does not exceed 2.

Abstract. The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential is a smooth function on a two-dimensional torus. It is known that the existence of integrals of degree 1 and 2 is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree n independent of the energy integral, then there exists an additional integral of degree 1 or 2. In the present paper this result is established for n = 3 (which generalizes a theorem of Byalyi), and for n = 4, 5, and 6, this is proved under some additional assumptions about the spectrum of the potential.

Abstract. Chaotic steady viscous flows are studied. The following two models are considered: a barotropic compressible viscous fluid and an incompressible ideal fluid with external friction. Based on the well-known Poincaré - Kozlov theorem, we show for these models that steady fluid flows are chaotic: nontrivial integrals, fields of symmetries, and integral invariants are absent.