  Two-particle discrete Schr\"{o}dinger operators $H(k)=H_{0}(k)-V$ on the
three-dimensional lattice $\Z^3,$ $k$ being the two-particle quasi-momentum,
are considered. An estimate for the number of the eigenvalues lying outside of
the band of $H_{0}(k)$ via the number of eigenvalues of the potential operator
$V$ bigger than the width of the band of $H_{0}(k)$ is obtained. The existence
of non negative eigenvalues below the band of $H_{0}(k)$ is proven for
nontrivial values of the quasi-momentum $k\in \T^3\equiv (-\pi,\pi]^3$,
provided that the operator H(0) has either a zero energy resonance or a zero
eigenvalue. It is shown that the operator $H(k), k\in \T^3,$ has infinitely
many eigenvalues accumulating at the bottom of the band from below if one of
the coordinates $k^{(j)},j=1,2,3,$ of $k\in \T^3$ is $\pi.$
