A question about equivariant closed form

Question

Let $M$ be a n-dimensional closed smooth manifold, $\eta_{k}$ be the k-form on $M$, $X$ be the smooth vector field on $M$.Let $\eta=\eta_{0}+\eta_{1}+\cdots+\eta_{n-1}+\eta_{n}$ be a ($d-i_{X}$)-closed form, then we have $$i_{X}\eta_{n}=d\eta_{n-2},$$ the relation imply that $\eta_{n}$ is exact outside the set $M_{0}$ of zeros of the vector field $X$. How to get the result?

Edit: I read this result in the chapter 7 of "Heat Kernels and Dirac Operators", page 203. My question is why the relation imply that $\eta_{n}$ is exact outside the set $M_{0}$.

Answer 1

If you can find a closed 1-form $\alpha\in \Omega^1(M\setminus Z(X))$ with $i_X\alpha=1$, then $\eta_n=\alpha\wedge i_X\eta_n = \alpha\wedge d\eta_{n-2}= -d(\alpha\wedge\eta_{n-2})$ is exact. Maybe, the source that you are reading has such $\alpha$.