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 $\alpha$.
Peter Michor
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