The total spine of the Milnor fibration of a plane curve singularity

For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the vector field $-\mathrm{\nabla }\log \mathrm{\mid }f\mathrm{\mid }$. Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, $S$, into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius $0$. The strict transform of $S$ in this space induces the aforementioned spine for each fiber of this fibration. These fibers are naturally endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary.