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In dimension 7, we establish a Fredholm theory for a Dirac-type operatorassociated to a connection with point singularities. There are twoapplications. $1$. over a closed 7-manifold, under some natural conditions, a$G_{2}-$instanton and its point singularities can still be "seen" when the$G_{2}-$structure is properly perturbed. $2$. over a ball in $\R^{7}$, for anyalmost-Euclidean $G_{2}-$structure, there exists a $G_{2}-$monopole asymptoticto an arbitrary Hermitian Yang-Mills connection on $S^{6}$.

Deformation of singular connections I: $G_{2}-$instantons with point singularities