  We study the Schroedinger operators H_{\lambda\mu}(K), with K \in T_2 the
fixed quasi-momentum of the particles pair, associated with a system of two
identical fermions on the two-dimensional lattice Z_2 with first and second
nearest-neighboring-site interactions of magnitudes \lambda \in R and \mu \in
R, respectively. We establish a partition of the (\lambda,\mu)-plane so that in
each its connected component, the Schroedinger operator H_{\lambda\mu}(0) has a
definite (fixed) number of eigenvalues, which are situated below the bottom of
the essential spectrum and above its top. Moreover, we establish a sharp lower
bound for the number of isolated eigenvalues of H_{\lambda\mu}(K) in each
connected component.
