Vector fields with simply connected trajectories transverse to a polynomial

We classify the polynomial vector fields X on C2 which have simply-connected trajectories and satisfy dP(X)=1 for P in C[x,y]. In particular, we obtain that such vector fields are complete and have all its trajectories proper and of type C. Finally, we apply this result to classify surjective C-derivations of C[x,y], and provide a proof of a conjecture of Cerveau [7].