  We study the existence of eigenvalues of the generalized Friedrichs model
$H_\mu(p)$, with a rank-one perturbation, depending on parameters $\mu>0$ and
$p\in\mathbb{T}^2$, and found an absolutely convergent expansions for
eigenvalues at $\mu(p)$, the coupling constant threshold. The expansions are
highly dependent on that, whether the threshold $m(p)$ of the essential
spectrum is: $(i)$ neither an threshold eigenvalue nor a threshold resonance;
$(ii)$ a threshold resonance; $(iii)$ an threshold eigenvalue.
