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Asymptotics of the monodromy transformation in certain classes of monodromy germs

We calculate the second term of the asymptotics of the monodromy transformation of a monodromic singular point of an analytic vector field on a plane whose Newton diagram consists of one or two edges. In the cases under consideration the principal term of the monodromy transformation coincides with the identity function. The case of two edges is characterized by the fact that, as a result of blowing up the singularity by the Newton diagram, a singular point emerges that is a degenerate saddle. The results obtained make it possible to state a sufficient condition for the existence of a focus and to construct the stability boundary in the classes of vector fields under consideration.