Let $M$ orientable closed smooth manifold. Let $\Omega$ volume form for $M$. A smooth $\Phi_t$ flow defined on $M$ is volume-preserving with respect to $\Omega$ if $\mathcal{L}_X\Omega=0,$ where $X$ is a vector field generating the flow. In this case, the flux form defined by $\omega=i_X\Omega$ is closed $(n-1)$-form ($\mathcal{L}_X$ denote the Lie derivative in the $X$ direction and $i_X$ the interior product). In fact, by the Cartan's formula we have $\mathcal{L}_X\Omega=di_X\Omega+i_Xd\Omega$. A cross section for a flow is a closed comdimension one smooth submanifold of $M$ that interset every line of the flow transversaly. If the flux form of a non singular volume-preserving flow is null in cohomology, this flow can't admit cross section (if $N$ is a cross section then $([\omega],[N])=\int _N\omega\neq 0$ with implies that $[\omega]\neq 0$ in $H^{n-1}(M;\mathbb{R})$). The question is:

Exists a non-singular volume-preserving flow with non exact flux form without cross section?