Separatrix lunes of analytic vector fields of the plane (Q1096221)

The theorems on sectorial normalization of resonance vector fields and mappings are given which can be applied to the investigation of the neighbourhood of sectorial biangular polygons. Theorem 1. Assume analytical vector fields on the real plane possess complicated cycles which consist of two elementary singular points and two separatrices - biangular separatrix polygon. Then in some neighbourhood of these complicated cycles there are no limit cycles. Theorem 2. The plane polynomial vector field with second order terms (quadratic field) possesses only a finite number of limit cycles.