On complex vector fields having simply-connected orbits

In this paper we study the classification polynomial vector fields with isolated singularities on C2 under the hypothesis that the non-singular orbits are simply-connected. Also we regard the case these orbits are cylinders. Regardless the natural relations with the study of complete vector fields on C2 which is carried out in [9], we give examples where the vector field is not complete. Our techniques are based on the geometry of the corresponding projective foliation.