I heard that there is a theorem due to Rosenlicht which says that:
  

Theorem:Let $X$ be a complex projective manifold,and $V$ is a given holomorphic vector field on $X$.Then $X$ is uniruled,ie,can be covered by rational curves,if $V$ has a zero.

I have thought for a few days and failed to give myself a proof.Can somebody give me the reference or say something about the idea of proof?

 Thanks in advance.

I heard that there is a theorem due to Rosenlicht which says the following:

Theorem. Let $X$ be a complex projective manifold and $V$ a non-trivial holomorphic vector field on $X$. Then $X$ is uniruled,ie,can be covered by rational curves,if $V$ has a zero.

I have thought for a few days and failed to give myself a proof. Can somebody give me the reference or say something about the idea of proof?

Thanks in advance.