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We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that thegeodesic flow admits a nontrivial integral quadratic in velocities. Weconstruct local normal forms of such metrics. We show that these metrics havecertain useful properties similar to those of Riemannian Liouville metrics,namely: 1) they admit geodesically equivalent metrics; 2) one can use them toconstruct a big family of natural systems admitting integrals quadratic inmomenta; 3) the integrability of such systems can be generalized to the quantumsetting; 4) these natural systems are integrable by quadratures.

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