  One of the various versions of the classical Lyapunov-Poincar\'e 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 (\cite{moussu}). 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 Poincar\'e-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.
