  In this paper we give some positive and negative results about the contact
property for the energy levels $\Sigma_c$ of a symplectic magnetic field on
$S^2$. In the first part we focus on the case of the area form on a surface of
revolution. We state a sufficient condition for an energy level to be of
contact type and give an example where the contact property fails. If the
magnetic curvature is positive, the dynamics and the action of invariant
measures can be numerically computed. This hints at the conjecture that an
energy level of a symplectic magnetic field with positive magnetic curvature
should be of contact type. In the second part we show that, for small energies,
there exists a convex hypersurface $N_c$ in $\mathbb C^2$ and a covering map
$p_c:N_c \rightarrow \Sigma_c$ such that the pull-back via $p_c$ of the
characteristic distribution on $\Sigma_c$ is the standard characteristic
distribution on $N_c$. As a corollary we prove that there are either two or
infinitely many periodic orbits on $\Sigma_c$. The second alternative holds if
there exists a contractible prime periodic orbit.
