On Rational Integrals of Geodesic Flows on 2-Surfaces

In this paper we study Riemannian metrics on 2-surfaces with integrable geodesic flows by means of an additional rational-in-momenta first integral. This problem is reduced to a quasi-linear system of PDEs. We construct solutions to this system via the classical hodograph method. These solutions give rise to local examples of metrics and rational integrals. Some of the constructed metrics have a very simple form. A family of implicit integrable examples parameterized by two arbitrary functions of one variable is also provided.