  It has been showed by Byde that it is possible to attach a Delaunay-type end
to a compact nondegenerate manifold of positive constant scalar curvature,
provided it is locally conformally flat in a neighborhood of the attaching
point. The resulting manifold is noncompact with the same constant scalar
curvature. The main goal of this paper is to generalize this result. We will
construct a one-parameter family of solutions to the positive singular Yamabe
problem for any compact non-degenerate manifold with Weyl tensor vanishing to
sufficiently high order at the singular point. If the dimension is at most 5,
no condition on the Weyl tensor is needed. We will use perturbation techniques
and gluing methods.
