Let $M$ be a contact manifold, and let $F$ be an oriented 1-dimensional foliation that is transverse to the contact structure. 


Is there a contact form $\alpha$ whose associated Reeb vector field generates the foliation $F$?



In a contact manifold, is every tranverse 1-foliation given by some Reeb vector field?
Answer is negative. A possible solution is to construct a transversal vector field without closed trajectories. By Taubes theorem (see his paper "The Seiberg–Witten equations and the Weinstein conjecture") it is impossible. One could easily prove that such a field exists for a standard $3$-torus with contact structure given by $cos\theta dx + sin \theta dy$.


