Surfaces with Central Configuration and Dulac’s Problem for a Three Dimensional Isolated Hopf Singularity

Let \(\xi \) $\xi$ be a real analytic vector field with an elementary isolated singularity at \(0\in \mathbb {R}^3\) $0\in {\mathbb{R}}^3$ and eigenvalues \(\pm bi,c\) $\pm bi,c$ with \(b,c\in \mathbb {R}\) $b,c\in \mathbb{R}$ and \(b\ne 0\) $b\ne 0$ . We prove that all cycles of \(\xi \) $\xi$ in a sufficiently small neighborhood of 0, if they exist, are contained in the union of finitely many subanalytic invariant surfaces, each one entirely composed of a continuum of cycles. In particular, we solve Dulac’s problem for such vector fields, i.e., finiteness of limit cycles.