The slow zero-sound mode expected near the Mott transition in strongly
interacting two-dimensional Fermi systems that are neutral is shown t
o persist as the physical sound mode in the case that the fermion carr
ies electronic charge and is embedded in a positive ionic background.
The latter sound velocity softens completely precisely at the Mott tra
nsition, indicating that a zone-center structural transition will occu
r in the system. We suggest that this phonon-softening mechanism is re
lated to the structural transitions commonly observed in the oxide sup
erconductors.