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In this paper, the sixth in series, we continue our analysis of the interplaybetween non-Fermi liquid and pairing in the effective low-energy model offermions with singular dynamical interaction $V(\Omega_m) = {\barg}^\gamma/|\Omega_m|^\gamma$ (the $\gamma$ model). The model describeslow-energy physics of various quantum-critical metallic systems at the verge ofan instability towards density or spin order, pairing of fermions at thehalf-filled Landau level, color superconductivity, and pairing in SYK-typemodels. In previous Papers I-V we analyzed the $\gamma$ model for $\gamma \leq2$ 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 windsup along real frequency axis due to emerging dynamical vortices in the upperhalf-plane of frequency, and that there is a quantum phase transition at$\gamma =2$, when the number of dynamical vortices becomes infinite. In thispaper we consider larger $2< \gamma <3$ and address the issue what happens onthe other side of this quantum transition. We argue that the system moves awayfrom criticality in that the number of dynamical vortices becomes finite anddecreases with increasing $\gamma$. The ground state is again a superconductor,however a highly unconventional one with a non-integrable singularity in thedensity of states at the lower edge of the continuum. This implies that thespectrum of excited states now contains a level with a macroscopic degeneracy,proportional to the total number of states in the system. We argue that thephase diagram in variables $(T,\gamma)$ contains two distinct superconductingphases for $\gamma <2$ and $\gamma >2$, and an intermediate pseudogap state ofpreformed pairs.

physical review. b./physical review. b

Interplay between superconductivity and non-Fermi liquid at a  quantum-critical point in a metal. VI. The $\gamma$ model and its phase  diagram at $2 < \gamma <3$