Moser's theorem state's that
Given a closed connected $n$-dimensionnal manifold $M$ and two volume forms $\alpha$ and $\beta$ with $\int_M\alpha = \int_M\beta$, then there exists a diffeomorphism $\phi$ such that $\phi^*\alpha = \beta$.
The proof goes as follows:
$\alpha-\beta$ is of integral $0$ so there exists a $\gamma\in\Omega^{n-1}(M)$ such that $d\gamma = \alpha-\beta$. (Now the part I don't get) There exists also a Vector field $X$ on $M$ such that $\text{int}(X)\alpha = -\gamma$ where $\text{int}()$ is the interior derivative.
Why does such a $X$ exist?