Abstract: For a given finite subset S of a compact Riemannian manifold (M; g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary andsufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of S corresponds to an asymptotically flat end andthat the Schouten tensor of the new conformal metric belongs to the boundary of the given cone.  This is a joint work with Luc Nguyen.