Consider a smooth nowhere-zero $1$-form $\alpha$ on a smooth manifold $M$. Assume that
$\alpha \wedge d\alpha \equiv 0$.
Does it follow that there exists a smooth nowhere-zero function $f:M\to \Bbb R$ such that $f\alpha$ is closed?
(By the Frobenius theorem, for every point $x\in M$ there is a neighborhood $U\ni x$ and smooth $f_U:M\to \Bbb R$ such that $f_U \alpha|_U$ is closed, i.e. the local answer to my question is "yes". I am asking the global question.)