We consider completely integrable vector fields, i.e. local holomorphic vector fields that possess a maximum number of independent first integrals. In particular we will focus in dimension 3. A priori a completely integrable vector field should be easy to understand since its trajectories are the levels of a holomorphic map but there are interesting open problems concerning its geometrical properties and the algebraic structure of its space of first integrals. We will show that a completely integrable vector field either has infinitely many holomorphic invariant curves through the origin or its singularity at the origin is not isolated. This generalizes a result by Pinheiro and Reis under much more restrictive hypotheses. Our proof is of geometrical type. This is a joint work with Felipe Cano and Marianna Ravara Vago.