Let $\Omega\subset \mathbb{C}$ be an open subset, $f\in C^\infty (\Omega)$ and $\alpha(z)= df= \partial_x f\, dx + \partial_y f\, dy$ be such that $\alpha(z)\neq 0,\forall\,\, z\in \Omega$. Let $\omega=R\, dx\wedge dy$ be a $C^\infty$ 2-form on $\Omega$. I want to find out all $C^\infty$ functions $P$ and $Q$ such that $$\alpha \wedge(P\, dx +Q\, dy)= \omega.$$

Now $P = \frac{-R\, \partial_x f}{(\partial_x f)^2 + (\partial_y f)^2}$ and $Q = \frac{R\, \partial_y f}{(\partial_x f)^2 + (\partial_y f)^2}$ is a solution, so are $(\tilde P, \tilde Q )= (P+ \lambda \partial_x f, Q+ \lambda\partial_y f),\,\, \lambda \in \mathbb{C}$.

Now my question is whether it is the complete list of the solutions. If yes, how to prove it. If not, how to find all solutions.

Thank you.


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