  In this paper we continue our analysis of the interplay between the pairing
and the non-Fermi liquid behavior in a metal for a set of quantum-critical
models with an effective dynamical electron-electron interaction $V(\Omega_m)
\propto 1/|\Omega_m|^\gamma$ (the $\gamma$-model). We analyze both the original
model and its extension, in which we introduce an extra parameter $N$ to
account for non-equal interactions in the particle-hole and particle-particle
channel. In two previous papers(arXiv:2004.13220 and arXiv:2006.02968), we
considered the case $0 < \gamma <1$ and argued that (i) at $T=0$, there exists
an infinite discrete set of topologically different gap functions, $\Delta_n
(\omega_m)$, all with the same spatial symmetry, and (ii) each $\Delta_n$
evolves with temperature and terminates at a particular $T_{p,n}$. In this
paper, we analyze how the system behavior changes between $\gamma <1$ and
$\gamma >1$, both at $T=0$ and a finite $T$. The limit $\gamma \to 1$ is
singular due to infra-red divergence of $\int d \omega_m V(\Omega_m)$, and the
system behavior is highly sensitive to how this limit is taken. We show that
for $N =1$, the divergencies in the gap equation cancel out, and $\Delta_n
(\omega_m)$ gradually evolve through $\gamma=1$ both at $T=0$ and a finite $T$.
For $N \neq 1$, divergent terms do not cancel, and a qualitatively new behavior
emerges for $\gamma >1$. Namely, the form of $\Delta_n (\omega_m)$ changes
qualitatively, and the spectrum of condensation energies, $E_{c,n}$ becomes
continuous at $T=0$. We introduce different extension of the model, which is
free from singularities for $\gamma >1$.
