reduces Theorem \[thm:contact\_sys\] to a generalised version of Proposition \[p:hamsys\] for surfaces with boundary, where $\omega$ is symplectic in the interior and vanishes of order one at the boundary.

To give a unified model for the above two situations, we notice that the Hamiltonian diffeomorphism $\varphi:M\to M$ can be seen as the return map of a flow $\Phi^X$ on the manifold $M\times S^1$. If $H:M\times S^1\to{{\mathbb R}}$ is a Hamiltonian one-periodic in time generating $\varphi$ as time one-map, then $X(q,t):={\partial}_t+X_{H_t}(q)$, for all $(q,t)\in M\times S^1$. Here, $X_{H_t}$ is the Hamiltonian vector field tangent to $M\times\{t\}$ associated with the function $H_t:=H(\cdot,t)$. Thus, we view Proposition \[p:hamsys\] as giving a systolic-diastolic inequality for those flows on the trivial $S^1$-bundle $\mathfrak p:M\times S^1\to M$, which are obtained by lifting a Hamiltonian isotopy of $M$. On the other hand, Theorem \[thm:contact\_sys\] yields a systolic-diastolic inequality for Reeb flows on the non-trivial $S^1$-bundles obtained from the Boothby-Wang construction. We will show that these two types of flows are particular incarnations of the flow induced by an odd-symplectic structure (also known as Hamiltonian structure, see [@CM05]) on an oriented odd-dimensional closed manifold which is the total space of an *arbitrary* oriented circle bundle over a symplectic 