Threshold phenomenon for a family of the Generalized Generalized Friedrichs models with the perturbation of rank one

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.