  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.
