Abstract
One of the various versions of the classical Lyapunov-Poincaré 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 démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 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 Poincaré-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.
Acknowledgment
The first author is grateful to Edital n°77/2022/PRPPG-PAAP-UNILA for partially supporting this research work.
Disclosure statement
No potential conflict of interest was reported by the author(s).