On local description of two-dimensional geodesic flows with a polynomial first integral

Abstract

In this paper we present a construction of multiparametric families of two-dimensional metrics with a polynomial first integral of arbitrary degree in momenta. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic-type system. We give a constructive algorithm for the solution of the derived hydrodynamic-type system, i.e. we found infinitely many conservation laws and commuting flows. Thus we were able to find infinitely many particular solutions of this hydrodynamic-type system by the generalized hodograph method. Therefore infinitely many particular two-dimensional metrics equipped with first integrals polynomial in momenta were constructed.