First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity of foci with pure imaginary eigenvalues and with homogeneous nonlinearities of arbitrary degree having either its radial or angular speed independent of the angle variable in polar coordinates. After we study the cyclicity of a 0cm 0cm 0pt" We present an alternative algorithm for computing Poincaré–Lyapunov constants of simple monodromic singularities of planar analytic vector ?elds based on the concept of inverse integrating factor. Simple monodromic singular points are those for which after performing the ?rst (generalized) polar blow-up, there appear no singular points. In other words, the associated Poincaré return map is analytic. We study the maximum number of limit cycles that can bifurcate from a focus singular point of an analytic , autonomous differential system in the real plane under an analytic perturbation . We consider being a focus singular point of the following three types: non-degenerate , degenerate without characteristic directions and nilpotent.