  We consider a family $$ \widehat H_{a,b}(\mu)=\widehat H_0 +\mu \widehat
V_{a,b}\quad \mu>0, $$ of Schr\"odinger-type operators on the two dimensional
lattice $\mathbb{Z}^2,$ where $\widehat H_0$ is a Laurent-Toeplitz-type
convolution operator with a given Hopping matrix $\hat{e}$ and $\widehat
V_{a,b}$ is a potential taking into account only the zero-range and one-range
interactions, i.e., a multiplication operator by a function $\hat v$ such that
$\hat v(0)=a,$ $\hat v(x)=b$ for $|x|=1$ and $\hat v(x)=0$ for $|x|\ge2,$ where
$a,b\in\mathbb{R}\setminus\{0\}.$ Under certain conditions on the regularity of
$\hat{e}$ we completely describe the discrete spectrum of $\hat H_{a,b}(\mu)$
lying above the essential spectrum and study the dependence of eigenvalues on
parameters $\mu,$ $a$ and $b.$ Moreover, we characterize the threshold
eigenfunctions and resonances.
