forms $\Omega$ with vanishing cohomology class are exactly the differentials of Zoll contact forms $\alpha$. We can use this observation to extend the classification of Zoll contact forms on three-manifolds contained in [@BK19a Proposition 1.2] to Zoll odd-symplectic forms.

\[p:cla\] Let $\Sigma$ be a connected oriented closed three-manifold. There is a Zoll odd-symplectic form on $\Sigma$ if and only if $\Sigma$ is the total space of an oriented $S^1$-bundle over a connected oriented closed surface $M$. When the bundle is non-trivial, then every Zoll odd-symplectic form is the differential of a Zoll contact form. When the bundle is trivial, then the group $H_1(\Sigma;{{\mathbb Z}})$ is free with rank equal to $2\, \mathrm{genus}(M)+1$, and if $\Omega$ and $\Omega'$ are Zoll odd-symplectic forms on $\Sigma$, there exists a real number $T>0$ and a diffeomorphism $\Psi:\Sigma\to\Sigma$ such that $\Psi^*\Omega'=T\Omega$. Moreover,

-   if $\Sigma=S^2\times S^1$, the set ${{\mathcal Z}}_C(\Sigma)$ has exactly two connected components, $\forall\, C\in H^2_{{{\mathrm{dR}}}}(\Sigma)\setminus 0$;

-   if $\Sigma={{\mathbb T}}^2\times S^1$, the set ${{\mathcal Z}}_C(\Sigma)$ is non-empty and connected, $\forall\, C\in H^2_{{{\mathrm{dR}}}}(\Sigma)\setminus 0$;

-   if $\Sigma=M\times S^1$ with $\chi(M)<0$, there is a one-dimensional subspace $L\subset H^2_{{{\mathrm{dR}}}}(\Sigma)$ such that ${{\mathcal Z}}_C(\Sigma)$ is non-empty if and only if $C\in L\setminus 0$; in this case 