The linear term of the Poincaré map at singularities of planar vector fields

The aim of this work is to give conditions on the parameters of a family of analytic planar vector fields with a fixed Newton diagram and a monodromic singularity in order to guarantee that the coefficient (generalized first Lyapunov constant) of the linear part of the Poincaré map has an explicit formula that depends only on the Newton diagram. As far as we know, all the monodromic classes appearing in the literature satisfy that formula.