  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: 1) they admit geodesically equivalent metrics; 2) one can use them to
construct a big family of natural systems admitting integrals quadratic in
momenta; 3) the integrability of such systems can be generalized to the quantum
setting; 4) these natural systems are integrable by quadratures.
