I have to construct a foliation $\mathfrak{F}$ of the 2-torus $T=S^1\times S^1$ such that some leaves of $\mathfrak{F}$ are compact while some other ones are not compact.
MY IDEA. Let's fix $\alpha$ irrational number. My idea is to use somehow the injective immersions given by $f_\eta:\mathbb{R}\to T, f_\eta(t)=(e^{it},e^{i(\alpha t+\eta)})$ with $\eta\in\mathbb{R}$. I just know that for all $\eta\in\mathbb{R}$ the image $f_\eta(\mathbb{R})$ is a dense subset of $T$, then it can't be compact. Can you help me please defining $\mathfrak{F}$?
To construct such an example, think of the torus as a vertical cylinder with top and bottom circles identified. Now consider a flow that has exactly one periodic orbit and any other orbit spirals down to the bottom circle.
It's straightforward that this comstruction can be modified to have a foliation with exactly $p$ periodic orbits.
Edit: Here are some humble pictures for the foliations I'm proposing. The main idea is to foliate the cylinder $S^1\times [0,1]$ and then identify $(\theta,0)\cong (\theta,1)$. The regularity of the foliation desired was not stated but this identification can be done in $C^\infty$ if needed (certainly doing it in $C^0$ is no problem): The following foliation of the cylinder gives a foliation of the torus with exactly one compact leaf. The said compact leaf is a generator of the fundamental group, and it is both the $\alpha$- and $\omega$-limit set of any leaf:
To make this more rigorous one can think of it as the integral curves of an ODE. The vector field could be chosen so that there is no change of speed of rotation, but as $z$ approaches $0$ or $1$ the vertical speed decays to zero. Stacking (finitely or countably many) copies of this and then identifying the topmost and the bottommost circles we obtain a foliation with the number of compact leaves being any predetermined number, e.g. for exactly three compact leaves one has: