  Recently, monatomic chains on surfaces have been synthesized which show
evidence of Luttinger liquid physics. The experimental data point to a
dispersion along the chain with four Fermi points. Here we investigate a
general low-energy effective Hamiltonian for such a two-band model where SU(2)
spin symmetry is broken but time reversal symmetry persists, as is expected due
to the surface geometry. Spin-orbit coupling gives rise to a new energy scale
epsilon_{SO} much smaller than the Fermi energy epsilon_F and to spin
non-conserving scattering processes. We derive the generic phase diagram at
zero temperature as well as an effective phase diagram at temperatures
epsilon_{SO} < T << epsilon_F. For the part of the phase diagram where a
Luttinger liquid is found to be stable, the density of states and the spectral
function are calculated and discussed in relation to the experimental data.
