On tensor invariants of dynamical systems on threedimentional manyfolds

V. V. Kozlov

We consider dynamical systems on compact threedimensional manifolds which have an invariant volume form. An important example is given by Hamilton equations of a system with two degrees of freedom restricted to three-dimensional lever surface of the energy integral. In this system we study the existence of tensor invariants (a first integral, a symmetry field, an invariant form) and give conditions of integrability by quadratures under the existence of a tensor invariant. We show that the infinite number of nondegenerate periodic trajectories and spliting of separatices obstruct the existence of nontrivial integral invariants analytical on the three-dimensional manifeld