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We consider a $2 \times 2$ operator matrix ${\mathcal A}_\mu,$ $\mu>0$related with the lattice systems describing two identical bosons and oneparticle, another nature in interactions, without conservation of the number ofparticles. We obtain an analogue of the Faddeev equation and its symmetricversion for the eigenfunctions of ${\mathcal A}_\mu$. We describe the newbranches of the essential spectrum of ${\mathcal A}_\mu$ via the spectrum of afamily of generalized Friedrichs models. It is established that the essentialspectrum of ${\mathcal A}_\mu$ consists the union of at most three boundedclosed intervals and their location is studied. For the critical value $\mu_0$of the coupling constant $\mu$ we establish the existence of infinitely manyeigenvalues, which are located in the both sides of the essential spectrum of${\mathcal A}_\mu$. In this case, an asymptotic formula for the discretespectrum of ${\mathcal A}_\mu$ is found.

Analysis of the spectrum of a $2\times 2$ operator matrix. Discrete spectrum asymptotics