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We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in{\Bbb T}^3:=(-\pi; \pi]^3$ associated with the lattice systems describing twoidentical bosons and one particle, another nature in interactions, withoutconservation of the number of particles. We find a finite set $\Lambda \subset{\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bfH}(K)$ for all $K \in \Lambda$ when the associated Friedrichs model has a zeroenergy resonance. It is found that for every $K \in \Lambda,$ the number $N(K,z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z,$ $z<0,$ satisfiesthe asymptotic relation $\lim\limits_{z \to -0} N(K, z)|\log|z||^{-1}={\mathcal U}_0$ with $0<{\mathcal U}_0<\infty,$ independently onthe cardinality of $\Lambda.$ Moreover, we prove that for any $K \in \Lambda$the operator ${\bf H}(K)$ has a finite number of negative eigenvalues if theassociated Friedrichs model has a zero eigenvalue or a zero is the regular typepoint for positive definite Friedrichs model.

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices