We consider a system of two arbitrary quantum particles on a two-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of operator family ℎ(?) depends on the particle interaction energy and the total quasi-momentum ? ∈ ? 2 (where ? 2 is a two-dimensional torus). Subject to the particle inter-action energy, we obtain conditions for existence of multiple eigenvalues below the essential spectrum of operator ℎ(?).