  We consider germs of holomorphic vector fields with an isolated singularity
at the origin $0\in\mathbb{C}^2$. We introduce a notion of stability, similar
to "Lyapunov stability". For such a germ, called $L$-stable singularity, either
the corresponding foliation admits a holomorphic first integral, or it is a
real logarithmic foliation singularity. A notion of $L$-stability is also
naturally introduced for a leaf of a foliation. In the complex codimension one
case, for holomorphic foliations, the holonomy groups of $L$-stable leaves are
proved to be abelian, of a suitable type. This implies the existence of local
closed meromorphic one-forms defining the foliation, in a neighborhood of
$L$-stable leaves.
