LAUSR

The geodesic flow (1.1) is called completely integrable, if there exists an additional first integral, i.e. a function F (x, p) such that dF dt = {F,H} ≡ 0, and F,H are functionally independent a.e.. Searching for metrics on 2-surfaces with integrable geodesic flows is a classical problem. There exist certain topological obstacles to the global integrability: on surfaces of any genus larger than 1 there are no analytic Riemannian metrics with analytically integrable geodesic flows (see [1]). Note that in the smooth case this result may be wrong in general (see [2], Chapter 3). In the overwhelming majority of known examples the first integrals are polynomials in momenta. The polynomial integrals of small degrees are well-studied (see, e.g., [3], [4]): the existence of a linear integral is related to the presence of a cyclic variable; the quadratic one is related to a possibility of the separation of variables. It was proved in [2], [5] that on 2-surfaces there exist Riemannian metrics whose geodesic flow admits a local polynomial integral of an arbitrarily high degree which is independent of the Hamiltonian. Rational in momenta first integrals of mechanical systems are also of an interest, they have been