# ordinary differential equations – Plane foliation with compact leaf must have a singularity

I’m trying to solve the following exercise from Camacho & Neto’s book $$textit{Geometric Theory of Foliations}$$:

How can I show that there is no compact leaf? I believe the idea is to suppose that there is such a compact leaf $$F$$, conclude that it must be diffeomorphic to $$S^{1}$$ and then use Poincaré-Bendixson theorem on the vector field associated to the line field $$P$$ defined by the foliation in order to find a singularity. However, PB theorem asks for some regularity on the vector field and, if we just construct the vector field this way, we might not have any sort of regularity.

I also believe that the idea at this point is to go to the orientable double covering of $$P$$ and there apply Poincaré-Bendixson, but I really don’t see how to do this. What am I missing?

Thanks in advance for any help!