About the Darboux theorem

Question

Let $w$ be a 2-form in the Novikov cohomology:

$$dw =\theta \wedge w$$

with $d\theta =0$, $\theta$ a fixed 1-form.

Have  we the Darboux theorem?

Does it exist a diffeomorphism locally such that $w$ is put in a standard form?

Answer 1

Locally, we have $\theta =df$, so that we can apply the Darboux theorem to $e^{-f}w$ which is closed and non-degenerated.