# Is time-1 map of a Hamiltonian vector field defined on a cylinder always twist?

Suppose I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed curves. I think that then the time-1 map of the flow will be twist (with integrability following from the integrability of $H(q,p)$), but how do I prove the twist property?

Moreover, could one relax the assumptions on the Hamiltonian (i.e. defined on a cylinder, and all orbits are periodic) and still retain the twist property?