**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?



Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function
Yes, $U=\{z:a<|z|<b\}$ ($a>0$) and $f(z)=i/\bar z$. Orbits are circles $\{|z|=c\}$.


