  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.
