Special Hamiltonian $S^1$-actions on symplectic 4-manifolds

In this paper we consider symplectic 4-manifolds $(M,\omega )$ with $c_1(M,\omega )=0$ which admit a Hamiltonian $S^1$-action together with an equivariant Maslov condition on orbits of the group action. We call such spaces {\em special Hamiltonian $S^1$-spaces}. It turns out that there are no compact special Hamiltonian $S^1$-spaces. We classify all exact special Hamiltonian $S^1$-spaces and show that all of them admit the structure of a Stein surface.