  We consider the family of $3 \times 3$ operator matrices $H(K),$ $K \in {\Bbb
T}^{\rm d}:=(-\pi; \pi]^{\rm d}$ arising in the spectral analysis of the energy
operator of the spin-boson model of radioactive decay with two bosons on the
torus ${\Bbb T}^{\rm d}.$ We obtain an analogue of the Faddeev equation for the
eigenfunctions of $H(K)$. An analytic description of the essential spectrum of
$H(K)$ is established. Further, it is shown that the essential spectrum of
$H(K)$ consists the union of at most three bounded closed intervals.
