Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

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We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: • they admit geodesically equivalent metrics; • one can use them to construct a large family of natural systems admitting integrals quadratic in momenta; • the integrability of such systems can be generalized to the quantum setting; • these natural systems are integrable by quadratures.