  In recent experiments on conductance of one-dimensional (1D) channels in
ultra-clean samples, a diverse set of plateaus were found at fractions of the
quantum of conductance in zero magnetic field. We consider a discrete model of
strongly interacting electrons in a clean 1D system where the current between
weak tunneling contacts is carried by fractionally charged solutions. While in
the spinless case conductance remains unaffected by the interaction, as is
typical for the strongly interacting clean 1D systems, we demonstrate that in
the spinful case the peak conductance takes fractional values that depend on
the filling factor of the 1D channel.
