  We consider a wide class of the two-particle Schr\"{o}dinger operators
$H_{\mu}(k)=H_{0}(k)+\mu V, \,\mu>0,$ with a fixed two-particle quasi-momentum
$k$ in the $d$ -dimensional torus $\mathbb{T}^d$, associated to the
Bose-Hubbard hamiltonian $H_{\mu}$ of a system of two identical
quantum-mechanical particles (bosons) on the $d$- dimensional hypercubic
lattice $\mathbb{Z}% ^d$ interacting via short-range pair potentials. We study
the existence of eigenvalues of $H_{\mu}(k)$ below the threshold of the
essential spectrum depending on the interaction energy $\mu>0$ and the
quasi-momentum $k\in \mathbb{T}^d$ of particles. We prove that the threshold
(bottom of the essential spectrum), as a singular point (a threshold resonance
or a threshold eigenvalue), creates eigenvalues below the essential spectrum
under perturbations of both the coupling constant $\mu>0$ and the
quasi-momentum $k$ of the particles. Moreover, we show that if the threshold is
a regular point, then it does not create any eigenvalues under small
perturbations of the coupling constant $\mu>0$ and the quasi-momentum $k$.
