  This paper is about the integrability of complex vector fields in dimension
three in a neighborhood of a singular point. More precisely, we study the
existence of holomorphic first integrals for isolated singularities of
holomorphic vector fields in complex dimension three, pursuing the discussion
started in \cite{CaSc2009}. Under generic conditions, we prove a topological
criteria for the existence of a holomorphic first integral. Our result may be
seen as a kind of Reeb stability result for the framework of vector fields
singularities in complex dimension three. As a consequence, we prove that, for
the class of singularities we consider, the existence of a holomorphic first
integral is invariant under topological equivalence.
