  We study the separatrices at the origin of the vector field $-\nabla \log
|f|$, where $f$ is any plane curve singularity. 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 a spine for each fiber of this
fibration. These fibers are endowed with a vector field in such a way that the
spine consists of trajectories which do not escape through the boundary.
