Stratification of three-dimensional real flows II: A generalization of Poincaré's planar sectorial decomposition

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 Poincaré, about the decomposition of the dynamics of an analytic planar vector field into {\em parabolic}, {\em elliptic} or {\em hyperbolic} invariant sectors.