  We study the spin-orbit interaction effects in a one-dimensional electron
system that result from the image charges in a nearby metallic gate. The
nontrivial property of the image-potential-induced spin-orbit interaction
(iSOI) is that it directly depends on the electron density because of which a
positive feedback arises between the electron density and the iSOI magnitude.
As a result, the system becomes unstable against the density fluctuations under
certain conditions. In addition, the iSOI contributes to the electron-electron
interaction giving rise to strong changes in electron correlations and
collective excitation spectra. We trace the evolution of the spectrum of the
collective excitations and their spin-charge structures with the change in the
iSOI parameter. One out of two collective modes softens as the iSOI amplitude
grows to become unstable at its critical value. Interestingly, this mode
evolves from a pure spin excitation to a pure charge one. At the critical point
its velocity turns to zero together with the charge stiffness.
