  Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold
equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point
set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in
\cite{L}, we defined a certain invariant of such spaces which consists of fixed
point data and twist type, and we divided the possible values of these
invariants into six ``types''. In this paper, we construct such manifolds with
these ``types''. As a consequence, we have a precise list of the values of
these invariants.
