  We consider a $2 \times 2$ operator matrix ${\mathcal A}_\mu,$ $\mu>0$
related with the lattice systems describing two identical bosons and one
particle, another nature in interactions, without conservation of the number of
particles. We obtain an analogue of the Faddeev equation and its symmetric
version for the eigenfunctions of ${\mathcal A}_\mu$. We describe the new
branches of the essential spectrum of ${\mathcal A}_\mu$ via the spectrum of a
family of generalized Friedrichs models. It is established that the essential
spectrum of ${\mathcal A}_\mu$ consists the union of at most three bounded
closed intervals and their location is studied. For the critical value $\mu_0$
of the coupling constant $\mu$ we establish the existence of infinitely many
eigenvalues, which are located in the both sides of the essential spectrum of
${\mathcal A}_\mu$. In this case, an asymptotic formula for the discrete
spectrum of ${\mathcal A}_\mu$ is found.
