  A model operator $H_\mu,$ $\mu>0$ associated to a system of three particles
on the three-dimensional lattice $ \mathbb{Z}^3$ that interact via nonlocal
pair potentials is considered. We study the case where the parameter function
$w$ has a special form with the non degenerate minimum at the $n, n>1$ points
of the six-dimensional torus $\mathbb{T}^6.$ If the associated Friedrichs model
has a zero energy resonance, then we prove that the operator $H_\mu$ has
infinitely many negative eigenvalues accumulating at zero and we obtain an
asymptotics for the number of eigenvalues of $H_\mu$ lying below $z,$ $z<0$ as
$z\to -0.$
