# Solution of a wedge equation

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.