  We show that including pairing and repulsion into the description of 1D
spinless fermions, as in the domain wall theory of commensurate melting or the
interacting Kitaev chain, leads, for strong enough repulsion, to a line of
critical points in the eight vertex universality class terminating floating
phases with emergent U(1) symmetry. For nearest-neighbor repulsion and pairing,
the variation of the critical exponents along the line that can be extracted
from Baxter's exact solution of the XYZ chain at $J_x=-J_z$ is fully confirmed
by extensive DMRG simulations of the entire phase diagram, and the qualitative
features of the phase diagram are shown to be independent of the precise form
of the interactions.
