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