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We analyze a class of quantum-critical models, in which momentum integrationand the selection of a particular pairing symmetry can be done explicitly, andthe competition between non-Fermi liquid and pairing can be analyzed within aneffective model with dynamical electron-electron interaction $V(\Omega_m)\sim1/|\Omega_m|^\gamma$ (the $\gamma$-model). In this paper, the first in theseries, we consider the case $T=0$ and $0<\gamma <1$. We argue that tendency topairing is stronger, and the ground state is a superconductor. We argue,however, that superconducting state is highly non-trivial as there exists adiscrete set of topologically distinct solutions for the pairing gap $\Delta_n(\omega_m)$ ($n = 0, 1, 2..., \infty$). All solutions have the same spatialpairing symmetry, but differ in the time domain: $\Delta_n (\omega_m)$ changessign $n$ times as a function of Matsubara frequency $\omega_m$. The $n =0$solution $\Delta_0 (\omega_m)$ is sign-preserving and tends to a finite valueat $\omega_m =0$, like in BCS theory. The $n = \infty$ solution corresponds toan infinitesimally small $\Delta (\omega_m)$. As a proof, we obtain the exactsolution of the linearized gap equation at $T=0$ on the entire frequency axisfor all $0<\gamma <1$, and an approximate solution of the non-linear gapequation.We argue that the presence of an infinite set of solutions opens up anew channel of gap fluctuations. We extend the analysis to the case where thepairing component of the interaction has additional factor $1/N$ and show thatthere exists a critical $N_{cr} >1$, above which superconductivity disappears,and the ground state becomes a non-Fermi liquid.We show that all solutionsdevelop simultaneously once $N$ gets smaller than $N_{cr}$.

physical review. b./physical review. b

Interplay between superconductivity and non-Fermi liquid at a  quantum-critical point in a metal. I: The $\gamma$-model and its phase  diagram at $T=0$. The case $0 < \gamma <1$