  Let $\xi$ be an analytic vector field in $\mathbb{R}^3$ with an isolated
singularity at the origin and having only hyperbolic singular points after a
reduction of singularities $\pi:M\to\mathbb{R}^3$. The union of the images by
$\pi$ of the local invariant manifolds at those hyperbolic points, denoted by
$\Lambda$, is composed of trajectories of $\xi$ accumulating to $0 \in
\mathbb{R}^3$. Assuming that there are no cycles nor polycycles on the divisor
of $\pi$, together with a Morse-Smale type property and a non-resonance
condition on the eigenvalues at these points, in this paper we prove the
existence of a fundamental system $\{V_n\}$ of neighborhoods well adapted for
the description of the local dynamics of $\xi$: the frontier $Fr(V_n)$ is
everywhere tangent to $\xi$ except around $Fr(V_n)\cap\Lambda$, where
transvesality is mandatory.
