  A family $H_\mu(p),$ $\mu>0,$ $p\in\T^3$ of the Generalized Firedrichs models
with the perturbation of rank one, associated to a system of two particles,
moving on the three dimensional lattice $\mathbb{\Z}^3,$ is considered. The
existence or absence of the unique eigenvalue of the operator $H_\mu(p)$ lying
outside the essential spectrum, depending on the values of $\mu>0$ and $p\in
U_{\delta}(p_{\,0})\subset\T^3$ is proven. Moreover, the analyticity of
associated eigenfunction is shown.
