  When the rank of the bundle is $\geq 2$, in a certain sense, we found an
essential obstruction for the gluing construction of $G_{2}-$instantons with
$1-$dimensional singularities. It involves the Atiyah classes generated by
contracting a vector in $\mathbb{C}^{3}$ with the curvature. Intuitively
speaking, the gluing does not work if the tangent connection at a component of
the $1-$dimensional singular locus is not the twisted Fubini-Study connection
on a twisted tangent bundle of $\mathbb{P}^{2}$. Particularly, it fails if the
rank of the bundle is $\geq 3$.
