  Let $X$ be a compact K\"ahler manifold and $S$ a subvariety of $X$ with
higher co-dimension. The aim is to study complete constant scalar curvature
K\"ahler metrics on non-compact K\"ahler manifold $X-S$ with
Poincar\'e--Mok--Yau asymptotic property (see Definition \ref{def}). In this
paper, the methods of Calabi's ansatz and the moment construction are used to
provide some special examples of such metrics.
