  Let $(M, \omega)$ be a 6-dimensional closed symplectic manifold with a
symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq
2$. Assume that $\omega$ is integral with a generalized moment map $\mu$. We
first prove that the action is Hamiltonian if and only if $b_2^+(M_{\red})=1$,
where $M_{\red}$ is any reduced space with respect to $\mu$. It means that if
the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. Secondly, we
focus on the case when the action is semifree and Hamiltonian. We prove that if
$M^{S^1}$ consists of surfaces, then the number $k$ of fixed surfaces with
positive genera is at most four. In particular, if the extremal fixed surfaces
are spheres, then $k$ is at most one. Finally, we prove that $k \neq 2$ and we
construct some examples of 6-dimensional semifree Hamiltonian $S^1$-manifolds
such that $M^{S^1}$ contains $k$ surfaces of positive genera for $k = 0$ and 4.
Examples with $k=1$ and 3 were given in \cite{L2}.
