We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.

1 aGhezzi, Roberta1 aRemizov, Alexey, O. uhttps://www.math.sissa.it/publication/class-vector-fields-discontinuity-divide-zero-type-and-its-applications