  We analyze the superconducting instabilities in the vicinity of the
quantum-critical point of an inversion symmetry breaking order. We first show
that the fluctuations of the inversion symmetry breaking order lead to two
degenerate superconducting (SC) instabilities, one in the $s$-wave channel, and
the other in a time-reversal invariant odd-parity pairing channel (the simplest
case being the same as the of $^3$He-B phase). Remarkably, we find that unlike
many well-known examples, the selection of the pairing symmetry of the
condensate is independent of the momentum-space structure of the collective
mode that mediates the pairing interaction. We found that this degeneracy is a
result of the existence of a conserved fermionic helicity, $\chi$, and the two
degenerate channels correspond to even and odd combinations of SC order
parameters with $\chi=\pm1$. As a result, the system has an enlarged symmetry
$U(1)\times U(1)$, with each $U(1)\times U(1)$ corresponding to one value of
the helicity $\chi$. Because of the enlarged symmetry, this system admits
exotic topological defects such as a fractional quantum vortex, which we show
has a Majorana zero mode bound at its core. We discuss how the enlarged
symmetry can be lifted by small perturbations, such as the Coulomb interaction
or Fermi surface splitting in the presence of broken inversion symmetry, and we
show that the resulting superconducting state can be topological or trivial
depending on parameters. The $U(1)\times U(1)$ symmetry is restored at the
phase boundary between the topological and trivial SC states, and allows for a
transition between topologically distinct SC phases without the vanishing of
the order parameter. We present a global phase diagram of the superconducting
states and discuss possible experimental implications.
