  In this paper, the sixth in series, we continue our analysis of the interplay
between non-Fermi liquid and pairing in the effective low-energy model of
fermions with singular dynamical interaction $V(\Omega_m) = {\bar
g}^\gamma/|\Omega_m|^\gamma$ (the $\gamma$ model). The model describes
low-energy physics of various quantum-critical metallic systems at the verge of
an instability towards density or spin order, pairing of fermions at the
half-filled Landau level, color superconductivity, and pairing in SYK-type
models. In previous Papers I-V we analyzed the $\gamma$ model for $\gamma \leq
2$ and argued that the ground state is an ordinary superconductor for $\gamma
<1$, a peculiar one for $1<\gamma <2$, when the phase of the gap function winds
up along real frequency axis due to emerging dynamical vortices in the upper
half-plane of frequency, and that there is a quantum phase transition at
$\gamma =2$, when the number of dynamical vortices becomes infinite. In this
paper we consider larger $2< \gamma <3$ and address the issue what happens on
the other side of this quantum transition. We argue that the system moves away
from criticality in that the number of dynamical vortices becomes finite and
decreases with increasing $\gamma$. The ground state is again a superconductor,
however a highly unconventional one with a non-integrable singularity in the
density of states at the lower edge of the continuum. This implies that the
spectrum of excited states now contains a level with a macroscopic degeneracy,
proportional to the total number of states in the system. We argue that the
phase diagram in variables $(T,\gamma)$ contains two distinct superconducting
phases for $\gamma <2$ and $\gamma >2$, and an intermediate pseudogap state of
preformed pairs.

Superconductivity;Strongly Correlated Electrons