Solutions to the singular ${\sigma }_2-$Yamabe problem with isolated singularities

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular ${\sigma }_k-$Yamabe problem asks for a complete metric $g$ on $M\mathrm{\backslash }X$ conformal to $g_0$ with constant ${\sigma }_k-$curvature. The ${\sigma }_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the Schouten tensor of a Riemannian metric. The main goal of this paper is to find solutions to the singular ${\sigma }_2-$Yamabe problem with isolated singularities in any compact non-degenerate manifold such that the Weyl tensor vanishing to sufficiently high order at the singular point. We will use perturbation techniques and gluing methods.