**Edit :** According to the comments of Michael Renardy and Christian Remling I revise the question as follows:

Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such that $X $ has a closed orbit and is in the form $X=f(\bar{z})$ where $f$ is a holomorphic function on $\overline{U}=\{\bar{z}\mid z\in U\}$?

**Added after the answer by Prof. Duchon:** Is there an example of such vector field with an Isochronous band of closed orbits?