SEPARATRICES FOR REAL ANALYTIC VECTOR FIELDS IN THE PLANE

Let X be a germ of real analytic vector field at (R2, 0) with an algebraically 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 nu 0(X) or the Milnor number mu 0 (X) is even, then X has a formal separatrix, that is, a formal invariant curve at 0 is an element of R2. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.2020 MATH. SUBJ. CLASS. 32S65, 37F75, 34Cxx, 14P15.