I was recently speaking with someone who works in Computational Chemistry and they mentioned that in a lot of the computational simulations they do, they have systems that are not symplectic but still integrable. I believe that this means symplectic in the formal sense, but for the record the computation was described to me in terms of Jacobians associated to Hamiltonian flows (Something along the line of the treatment in the Wikipedia article on Symplectic Integrators). The system still has a well-defined non-degenerate bilinear form, but it may not be closed (i.e. an Almost Symplectic Structure). From his examples, I believe that Liouville's Theorem holds for a slightly bigger subcategory of the category of manifolds than the category of symplectic manifolds. I was just wondering what the most general conditions are for having a $2n$-dimensional smooth manifold with global, non-degenerate $2$-form $\omega$ satisfy Liouville's Theorem. I found this paper and I was just wondering if these are the most general conditions for such an integrable system.

Thanks!

stronglyHamiltonian as requiring additionally $i_X d\omega = 0$. If $X$ is a strongly Hamiltonian vector field, by Cartan's formula, $L_X \omega = 0$, which then implies $L_X \omega^n = 0$ (Liouville's theorem). What kind of generalization are you looking for? $\endgroup$ – Sam Lisi Apr 8 '11 at 23:47