We study the family Hγλμη(K) , K∈T2,  of discrete Schrödinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimensional lattice Z2,  interacting through on one site, nearest-neighbour sites and next-nearest-neighbour sites with interaction magnitudes γ∈R,λ∈R,μ∈R    and η∈R   respectively. We prove there existence an important invariant subspace of operator Hγλμη(0)  such that the restriction of the operator Hγλμη(0)  on this subspace has at most one eigenvalue  lying both as below the essential spectrum as well as above it, depending on the interaction magnitude η∈R  (only). We also give a sharp lower bound for the number of eigenvalues of Hγλμη(K) .