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 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.