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On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics. (English) Zbl 1284.37016
Summary: We study phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. Also we assume 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 applications in differential geometry: singularities of geodesic flows in various singular metrics on two-dimensional manifolds.

##### MSC:
 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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