  For a wide class of two-body energy operators $h(k)$ on the three-dimensional
lattice $\bbZ^3$, $k$ being the two-particle quasi-momentum, we prove that if
the following two assumptions (i) and (ii) are satisfied, then for all
nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its
threshold is non-empty. The assumptions are:
  (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of
the quasi-momentum has either an eigenvalue or a virtual level at the bottom of
its essential spectrum and (ii) the one-particle free
  Hamiltonians in the coordinate representation generate positivity preserving
semi-groups.
