T}}^*N)\ar[dl]^{\mathfrak p_J}.\\
        &N&}$$ Therefore, $\Omega_{g,\sigma}:=\mathfrak i_g^*({{\mathrm d}}\lambda_\mathrm{can}+\mathfrak p_{{{\mathrm T}}^*N}^*\sigma)$ is an odd-symplectic two-form on $S^*_gN$ with cohomology class $C:=\mathfrak p_g^*[\sigma]$. The class $C$ vanishes if and only if $N$ is a surface different from the two-torus. In general, it is well-known that there exists a Zoll form $\mathfrak p^*{\mathrm{\omega}}_\sigma$ in the class $C$. More specifically, ${\mathrm{\omega}}_\sigma$ is the symplectic form on $\mathbb P_{{\mathbb C}}({{\mathrm T}}^*N)$ defined as $${\mathrm{\omega}}_\sigma:=a\,{\mathrm{\omega}}_\mathrm{FS}+ \mathfrak p_J^*\sigma,$$ for some $a>0$ small enough (see [@Voi Proposition 3.18] when $(N,\sigma,J)$ is Kähler). Here, $\mathfrak p^*{\mathrm{\omega}}_\mathrm{FS}={{\mathrm d}}\eta$, where $\eta$ is a connection form for $\mathfrak p$, which, for all $x\in N$, restricts on the fibre $\mathfrak p_g^{-1}(x)\cong S^{2m-1}$ to the standard contact form on the sphere. The $\mathfrak p$-fibres are almost tangent to the characteristic distribution of $\Omega_{g,\sigma}$, if $\sigma$ is very big. However, the form $\Omega_{g,\sigma}$ and $\mathfrak p^*{\mathrm{\omega}}_\sigma$ are remarkably not close to each other, if $m>1$.

The relevance of this example stems from the fact that the characteristics of $\Omega_{g,\sigma}$ are the tangent lifts of the magnetic geodesics of the pair $(g,\sigma)$. A curve $c:{{\mathbb R}}\to N$ is called a magnetic geodesic if $g(\dot c,\dot c)\equiv 1$ and there holds $$\nabla^g_{\dot c}\dot c=-fJ\dot c,$$ where $\nabla^g$ is 