  We consider a $2\times2$ block operator matrix ${\mathcal A}_\mu$ $($$\mu>0$
is a coupling constant$)$ acting in the direct sum of one- and two-particle
subspaces of a bosonic Fock space. The location of the essential spectrum of
${\mathcal A}_\mu$ is described and its bounds are estimated. It is shown that
there exist the critical values $\mu_l^0(\gamma)$ with $\gamma>0$ and
$\mu_r^0(\gamma)$ with $\gamma<12$ of the coupling constant $\mu>0$ such that
for all $\gamma>0$ $(\gamma<12)$ the operator ${\mathcal A}_\mu$ with
$\mu=\mu_l^0(\gamma)$ $(\mu=\mu_r^0(\gamma)$ has infinitely many eigenvalues on
the l.h.s. $($r.h.s.$)$ of the its essential spectrum. We prove that for all
$\mu \not\in \{\mu_l^0(\gamma),\mu_r^0(\gamma)\}$ the operator ${\mathcal
A}_\mu$ has finitely many discrete eigenvalues on the l.h.s. and r.h.s. of its
essential spectrum.
