On $\mathbb{R}^2$, say, given an arbitrary 1-form $u = u_x dx + u_y dy$, does there exist a (non-zero) function $f(x,y)$ such that $f u$ is an closed 1-form, i.e. $$ \partial_x(f u_y) = \partial_y (f u_x) \ \ ? $$
This is a 1st order linear PDE for $f$. I guess given the appropriate smoothness conditions, such an equation always has solutions. Can they be explicitly constructed in terms of $u$?
[Local existence/construction of $f$ is enough.]