Separatrices for Real Analytic Vector Fields in the Plane

Let X be a germ of real analytic vector field at (R 2 , 0) with an algebracally isolated singularity.We say that X is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities.In this case, we prove that if either the order ν0(X) or the Milnor number µ0(X) is even, then X has a formal separatrix, that is, a formal invariant curve at 0 ∈ R 2 .This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.