On Local Description of Two-Dimensional Geodesic Flows with a Polynomial First Integral

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows us to present infinitely many particular metrics by the generalized hodograph method.