  We extend the results and methods of \cite{MP} to prove the existence of
constant positive scalar curvature metrics $g$ which are complete and conformal
to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a
disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. The
existence of solutions with isolated singularities occupies the majority of the
paper; their existence was previously established by Schoen \cite{S}, but the
proof we give here, based on the techniques of \cite{MP}, is more direct, and
provides more information about their geometry. When $\Lambda$ is discrete we
also establish that these solutions are smooth points in the moduli spaces of
all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}

Differential Geometry;Differential Geometry