On real center singularities of complex vector fields on surfaces

One of the various versions of the classical Lyapunov-Poincare center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une demonstration geometrique d'un theoreme de Lyapunov-Poincare, Asterisque 98-99 (1982), pp. 216-223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with 'many' periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincare-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.