  A model operator $H$ corresponding to a three-particle discrete Schr\"odinger
operator on a lattice $\Z^3$ is studied. The essential spectrum is described
via the spectrum of two Friedrichs models with parameters $h_\alpha(p),$
$\alpha=1,2,$ $p \in \T^3=(-\pi,\pi]^3.$ The following results are proven:
  1) The operator $H$ has a finite number of eigenvalues lying below the bottom
of the essential spectrum in any of the following cases: (i) both operators
$h_\alpha(0), \alpha=1,2,$ have a zero eigenvalue; (ii) either $h_1(0)$ or
$h_2(0)$ has a zero eigenvalue.
  2) The operator $H$ has infinitely many eigenvalues lying below the bottom
and accumulating at the bottom of the essential spectrum, if both operators
$h_\alpha(0),\alpha=1,2,$ have a zero energy resonance.
