  We consider a family of $2 \times 2$ operator matrices ${\mathcal A}_\mu(k),$
$k \in {\Bbb T}^3:=(-\pi, \pi]^3,$ $\mu>0$, acting in the direct sum of zero-
and one-particle subspaces of a Fock space. It is associated with the
Hamiltonian of a system consisting of at most two particles on a
three-dimensional lattice ${\Bbb Z}^3,$ interacting via annihilation and
creation operators. We find a set $\Lambda:=\{k^{(1)},...,k^{(8)}\} \subset
{\Bbb T}^3$ and a critical value of the coupling constant $\mu$ to establish
necessary and sufficient conditions for either $z=0=\min\limits_{k\in {\Bbb
T}^3} \sigma_{\rm ess}({\mathcal A}_\mu(k))$ ( or $z=27/2=\max\limits_{k\in
{\Bbb T}^3} \sigma_{\rm ess}({\mathcal A}_\mu(k))$ is a threshold eigenvalue or
a virtual level of ${\mathcal A}_\mu(k^{(i)})$ for some $k^{(i)} \in \Lambda.$
