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This is the first step in an attempt at a deformation theory for$G_{2}-$instantons with $1-$dimensional conic singularities. Under a set ofmodel data, the linearization yields a self-adjoint first order ellipticoperator $P$ on a certain bundle over $\mathbb{S}^{5}$. As a dimensionreduction, the operator $P$ also arises from Hermitian Yang-Mills connectionswith isolated conic singularities on a Calabi-Yau $3$-fold.  Using the Quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$,we describe the set of all eigenvalues of $P$ (denoted by $Spec P$). We showthat $SpecP$ consists of finitely many integers induced by certain sheafcohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced bythe spectrum of the rough Laplacian on the pullback endomorphism bundle over$\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can bedescribed fairly explicitly.  Using the representation theory of $SU(3)$ and the subgroup $S[U(1)\timesU(2)]$, we show an example in which $SpecP$ and the multiplicities can becompletely determined.

The spectrum of an operator associated with $G_{2}-$instantons with  $1-$dimensional singularities and Hermitian Yang-Mills connections with  isolated singularities