For a manifold $M$, let $f \in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that $\{{k.df: k \in C^{k}(M)\}}$ forms an integrable regular Pfaffian system, and that the level sets of $f$ form the integral manifolds of such a Pfaffian system.
The converse, I know, is not true. The best one has is the following: Given a Pfaffian system generated by a one-form $\alpha$ that is integrable, there exists an open nbd. $U$ for every $m \in M$ on which there exists $f, g \in C^k(U)$ s.t. $g.\alpha |_U=df$ where $g$ is nowhere zero, and $f^{-1}(t_0)$ corresponds to the integral manifolds restricted to $U$.
Yet I still see physics references talking about a global level set prescription for such a manifold $M$. Does someone know any simple example where the manifold has smooth foliations from integrability, but lacks a level set prescription?