  This is the first step in an attempt at a deformation theory for
$G_{2}-$instantons with $1-$dimensional conic singularities. Under a set of
model data, the linearization yields a self-adjoint first order elliptic
operator $P$ on a certain bundle over $\mathbb{S}^{5}$. As a dimension
reduction, the operator $P$ also arises from Hermitian Yang-Mills connections
with 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 show
that $SpecP$ consists of finitely many integers induced by certain sheaf
cohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced by
the spectrum of the rough Laplacian on the pullback endomorphism bundle over
$\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be
described fairly explicitly.
  Using the representation theory of $SU(3)$ and the subgroup $S[U(1)\times
U(2)]$, we show an example in which $SpecP$ and the multiplicities can be
completely determined.
