is one-dimensional. They are of special importance, as their closed characteristics are the periodic orbits of a flow.

A two-form $\Omega$ on $\Sigma$ is said to be **odd-symplectic**, if it is closed and maximally non-degenerate, namely if its characteristic distribution is an (oriented) real line bundle over $\Sigma$. We write $\mathcal S_C(\Sigma)$ for the subset of all odd-symplectic forms in $\Xi^2_C(\Sigma)$.

In $\mathcal S_C(\Sigma)$ we hope to find elements whose characteristic distribution is as simple as possible. To this purpose we introduce the set of oriented $S^1$-bundles with total space $\Sigma$: $$\mathfrak P(\Sigma):=\big\{\mathfrak p:\Sigma\to M\ \big|\ \mathfrak p\ \text{is an oriented $S^1$-bundle}\big\}.$$

An odd-symplectic form $\Omega$ is said to be **Zoll** if the oriented leaves of its characteristic distribution are the fibres of some $\mathfrak p_\Omega:\Sigma\to M_\Omega$ in $\mathfrak P(\Sigma)$. We write ${{\mathcal Z}}_C(\Sigma)$ for the set of Zoll forms in $\mathcal S_C(\Sigma)$.

Let $(N_1,\sigma_1)$ and $(N_2,\sigma_2)$ be two connected closed symplectic manifolds and suppose that $[\sigma_1]\in H^2_{{\mathrm{dR}}}(N_1)$ is an integral cohomology class. We build the product symplectic manifold $(N_1\times N_2,\sigma_1\oplus\sigma_2)$ and consider the oriented $S^1$-bundle $\mathfrak p:\Sigma\to N_1\times N_2$ whose Euler class is $-[\sigma_1\oplus 0]$. Then, $\Omega:=\mathfrak p^*(\sigma_1\oplus\sigma_2)$ is a Zoll form with cohomology class $C:=\mathfrak p^*[0\oplus\sigma_2]$.

Odd-symplectic Zoll 