On the integrability of holomorphic vector fields

We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal{F}(X)$ induced by a generic vector field $X$ on $({\mathbb{C}}^3,0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F\text{\hspace{0.17em}⁣}:({\mathbb{C}}^3,0)⟶({\mathbb{C}}^2,0)$ such that the leaves of $\mathcal{F}(X)$ are contained in the level curves of $F$.