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A criterion in terms of differential invariants for a metric on a surface tobe Liouville is established. Moreover, in this paper we completely solve ininvariant terms the local mobility problem of a 2D metric, considered byDarboux: How many quadratic in momenta integrals does the geodesic flow of agiven metric possess? The method is also applied to recognition of otherpolynomial integrals of geodesic flows.

Invariant characterization of Liouville metrics and polynomial integrals