Integral invariants and limit sets of planar vector fields

We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact α- or ω-limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then V vanishes at at least one point of K (and on all of K if V is real analytic or Morse).