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 PoincarĂ©-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 PoincarĂ©-Bendixson, but I really don’t see how to do this. What am I missing?

Thanks in advance for any help!