Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions II

Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the maximal and the minimal fixed component are both two dimensional, then $(M,\omega_M)$ is $S^1$-equivariantly symplectomorphic to some K\"{a}hler Fano manifold $(X, \omega_X, J)$ equipped with a certain holomorphic Hamiltonian $S^1$-action. We also give a complete list of all such Fano manifolds together with an explicit description of the corresponding $S^1$-actions.