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

TL;DR

This paper proves that for polynomial vector fields on C2 with a proper, non-algebraic C* trajectory, all trajectories are proper and mostly of type C*, providing an analytic version of Lin-Zaidenberg Theorem.

Contribution

It establishes a new characterization of trajectories of polynomial vector fields with a specific proper C* trajectory, extending the Lin-Zaidenberg Theorem analytically.

Findings

All trajectories are proper except possibly one.

Most trajectories are of type C*.

An analytic version of Lin-Zaidenberg Theorem is derived.

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.