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

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct (Theorem 1) 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 (Theorem 2); • one can use them to construct a large family of natural systems admitting integrals quadratic in momenta (Theorem 4); • the integrability of such systems can be generalized to the quantum setting (Theorem 5); • these natural systems are integrable by quadratures (Section 2.2.2).