Consider a foliation of $\mathbb{R}^2$, say coming from the trajectories of a vector field $X$. Its orbit space (the quotient of $\mathbb{R}^2$ by the relation "lying on the same trajectory") is seldom Hausdorff. Such foliated structures have been intensively studied, and a complete $C^r$-classification is due to Haefliger and Reeb in the case where $X$ is regular on a simply connected region (thanks to the same topological niceness of the plane used in the proof of Bendixon-Poincaré's theorem). Almost any reasonable one-dimensional, simply-connected non-Hausdorff manifold can be realized as the orbit space of a foliation.

The two main sources of non-separability of orbits are:

- Saddle singularities: the stable and unstable (half-)manifolds cannot be separated.
- Limit cycles: the limit cycle cannot be separated from the accumulating trajectories.

I believe that orbits space coming from *real-analytic* foliations should have a "nicer" structure. I expect also that the work of Kaplan, Haefliger, Reeb dating back from the 40--50's should have been generalized to the analytic setting. Is that so? Is there any special structure / characterization on the (non-Hausdorff) analytic orbits space of a real-analytic planar foliation that I should be aware of (and where can I find it)?

A special case of particular interest is where the the vector field $X$ is the realification of a holomorphic vector field on $\mathbb C\simeq \mathbb R^2$. Now there are no limit cycles. The topology of the phase-portrait looks simpler and the orbit space also. Is there any known characterization of the analytic one-dimensional (non-Hausdorff) manifolds that can arise in this very special case?