$\kappa\in\Xi^2_{e_0}(M_0)$. We consequently choose the form ${\mathrm{\omega}}_0$ so that ${\mathrm{\omega}}_*={\mathrm{\omega}}_0+A_*\kappa$, where $A_*=\mathcal A(\Omega_*)$. We say that a form $\Omega\in\Xi^2_C(\Sigma)$ is an **H-form**, if there exists a function $H:\Sigma\to{{\mathbb R}}$ such that $$\Omega=\Omega_0+{{\mathrm d}}(H\eta).$$ We call $H$ a defining Hamiltonian for $\Omega$. If $H$ is $C^0$-close to $A_*$, then $\Omega$ is odd-symplectic, and if $H$ is $C^k$-close to $A_*$, then $\Omega$ is $C^{k-1}$-close to $\Omega_*$. An H-form is called **quasi-autonomous** if there exist $q_{\min},q_{\max}\in M_0$ and defining Hamiltonians $H_{\min}$ and $H_{\max}$ such that $$\min_{\Sigma} H_{\min}=H_{\min}(z),\quad\forall\,z\in\mathfrak p^{-1}_0(q_{\min}),\qquad \max_{\Sigma} H_{\max}=H_{\max}(z),\quad\forall\,z\in\mathfrak p^{-1}_0(q_{\max}).$$ We say that $H$ is quasi-autonomous, if the corresponding $\Omega$ is quasi-autonomous. We establish Conjecture \[con:lsios\] for quasi-autonomous forms close to a Zoll odd-symplectic form.

\[p:qaos\] There exists a $C^2$-neighbourhood $\mathcal H$ of the constant $A_*$ in the space of quasi-autonomous functions over $\Sigma$ such that if $H\in\mathcal H$ and $\Omega=\Omega_*+{{\mathrm d}}(H\eta)$ then $$P\big(\mathcal A_{\min}(\Omega)\big)\leq{{\mathfrak{Vol}}}(\Omega)\leq P\big(\mathcal A_{\max}(\Omega)\big),$$ and any of the two equalities holds if and only if $\Omega$ is Zoll. In the Zoll case, $H$ is invariant under the holonomy of $\eta$. In particular, it is constant if $e_0\neq0$.

Let $\mathfrak p_0:M_0\times S^1\to M_0$ be the trivial bundle and take $\eta={{\mathrm d}}t$, where $t \in S^1$ is the angular coordinate. In 