Invariant Curves and Topological Invariants for Real Plane Analytic Vector Fields

Let Z be a germ of a singular real analytic vector field at O R2. We give conditions on the multiplicity and the Milnor number of Z which imply that the foliation defined by Z has a characteristic orbit or an analytic invariant curve with the hypothesis that Z is a real generalized curve. Then it is proved that for a non-dicritical real generalized curve, the multiplicity mod 2 is invariant under bilipshitz homeomorphisms preserving foliations