  Inelastic and elastic neutron scattering have been used to study a single
crystal of the Ni$_{54}$Mn$_{23}$Al$_{23}$ Heusler alloy over a broad
temperature range. The paper reports the first experimental determination of
the low-lying phonon dispersion curves for this alloy system. We find that the
frequencies of the TA$_2$ modes are relatively low. This branch exhibits an
anomaly (dip) at a wave number $\xi_{0} ={1/3}\approx 0.33$, which softens with
decreasing temperature. Associated with this anomalous dip at $\xi_{0}$, an
elastic central peak scattering is also present. We have also observed
satellites due to the magnetic ordering.
