  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 $\pi:M\to\mathbb{R}^3$. Assuming certain conditions
to be specified throughout the work at hand, we establish a theorem of
stratification of the dynamics of $\xi$ that generalizes to dimension three the
classical one, coming from Poincar\'{e}, about the decomposition of the
dynamics of an analytic planar vector field into {\em parabolic}, {\em
elliptic} or {\em hyperbolic} invariant sectors.
