  We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic
flows admit nontrivial integrals quadratic in momenta. As an application, we
solve the Beltrami problem on closed surfaces, prove the nonexistence of
quadratically-superintegrable metrics of nonconstant curvature on closed
surfaces, and prove the two-dimensional pseudo-Riemannian version of the
projective Obata conjecture.
