  We report first principles calculations of the structural parameters and
phonon dispersion of the tetragonal non-centrosymmetric heavy fermion compound
CeAuAl$_3$. Taking into account weak magnetoelastic interactions of the
rare-earth (RE) ions with the spectrum of phonons, we obtain an analytical
expression for the hybridization of quadrupole excitations and phonons from the
poles of the one-phonon Green-function. In the paramagnetic phase, we predict
the formation of mixed modes that may be observed by inelastic neutron
scattering. Our results show that magnetoelastic interactions, albeit being
moderate, play an important role in CeAuAl$_3$. This suggests that
magnetoelastic interactions may be equally important in a wide range of related
compounds.
