Complete classification of typical singularities of geodesic flows on 2-surfaces with pseudo-Riemannian metrics

In this article, the authors investigate the geodesic flow on a two-dimensional surface equipped with a sub-Riemannian metric of non-constant signature for which the singularities lie on a regular curve. The classification of such flows has been considered in previous papers [N. Pavlova and A.~O. Remizov, Uspekhi Mat. Nauk {bf 66} (2011), no.~6(402), 193--194; [msn] MR2963454 [/msn]; A.~O. Remizov and F. Tari, Geom. Dedicata {bf 185} (2016), 131--153; [msn] MR3570306 [/msn]], and the authors here work to complete the last remaining case: that of a non-isotropic singular point of a sub-Riemannian metric. Beginning at this point and going to the Lorentzian domain, there is a one-parameter family of geodesics with the same initial tangent direction and a semi-cubic parabola with a different initial tangent vector.