Proper trajectories of type C* of a polynomial vector field on C2

Abstract

We prove that if a polynomial vector field on C2 has a proper and non-algebraic trajectory analytically isomorphic to C* all its trajectories are proper, and except at most one which is contained in an algebraic curve of type C all of them are of type C*. As corollary we obtain an analytic version of Lin-Zaidenberg Theorem for polynomial foliations.