Let $$\xi $$
ξ
be a real analytic vector field with an elementary isolated singularity at $$0\in \mathbb {R}^3$$
0
∈
R
3
and eigenvalues $$\pm bi,c$$
±
b
i
,
c
with $$b,c\in \mathbb {R}$$
b
,
c
∈
R
and $$b\ne 0$$
b
≠
0
. We prove that all cycles of $$\xi $$
ξ
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.