Invariant characterization of Liouville metrics and polynomial integrals

In this paper a criterion for a metric on a surface to be Liouville is established, and it is given in terms of differential invariants of the metric. Moreover, here we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals the geodesic flow of a given metric possesses? The method is also applied to recognition of higher degree polynomial integrals of geodesic flows.