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.
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