  In this paper we produce families of complete, non-compact Riemannian metrics
with positive constant $\sigma_2$--curvature on the sphere $\mathbb S^n$,
$n>4$, with a prescribed singular set $\Lambda$ given by a disjoint union of
closed submanifolds whose dimension is positive and strictly less than
$(n-\sqrt{n}-2)/2$. The $\sigma_2$--curvature in conformal geometry is defined
as the second elementary symmetric polynomial of the eigenvalues of the
Schouten tensor, which yields a fully non-linear PDE for the conformal factor.
We show that the classical gluing method of Mazzeo-Pacard (JDG 1996) for the
scalar curvature still works in the fully non-linear setting. This is a
consequence of the conformal properties of the $\sigma_2$ equation, which imply
that the linearized operator has good mapping properties in weighted spaces.
Our method could be potentially generalized to any $\sigma_k$, $2\leq k<n/2$,
nevertheless, the numerology becomes too involved.
