  The Rice-Mele model has two topological and spatially-inversion symmetric
phases, namely the Su-Schrieffer-Heeger (SSH) phase with alternating hopping
only, and the charge-density-wave (CDW) phase with alternating energies only.
The chiral symmetry of the SSH phase is robust in position space, so that it is
preserved in the presence of the ends of a finite system and of textures in the
alternating hopping. However, the chiral symmetry of the CDW wave phase is
nonsymmorphic, resulting in a breaking of the bulk topology by an end or a
texture in the alternating energies. We consider the presence of solitons
(textures in position space separating two degenerate ground states) in finite
systems with open boundary conditions. We identify the parameter range under
which an atomically-sharp soliton in the CDW phase supports a localized state
which lies within the band gap, and we calculate the expectation value $p_y$ of
the nonsymmorphic chiral operator for this state, and the soliton electric
charge. As the spatial extent of the soliton increases beyond the atomic limit,
the energy level approaches zero exponentially quickly or inversely
proportionally to the width, depending on microscopic details of the soliton
texture. In both cases, the difference of $p_y$ from one is inversely
proportional to the soliton width, while the charge is independent of the
width. We investigate the robustness of the soliton level in the presence of
disorder and sample-to-sample parameter variations, comparing with a single
soliton level in the SSH phase with an odd number of sites.
