  We study a two-dimensional Hubbard-Holstein model with phonons treated in the
adiabatic limit. A Hartree-Fock decomposition is employed for the Hubbard term.
A range of electronic densities are discussed with special emphasis on the
quarter-filling ($n=0.5$). We argue that the quarter-filled system is relevant
for the electronic properties observed at the interface between LaAlO$_3$ and
SrTiO$_3$, where half-electron per unit cell is transferred to the TiO$_2$
layer as a consequence of the polar discontinuity at the interface. In addition
to presenting the overall phase diagrams, we identify an interesting
charge-ordered antiferromagnetic phase for $n=0.5$, which was also reported
recently in the ab-initio study of the LaAlO$_3$-SrTiO$_3$ interface.
