  We conducted an investigation on the temperature and magnetic field
dependence of the elastic properties of UCoAl. The longitudinal elastic
stiffness, $C_{33}$, exhibits significant softening as the system approaches
the critical endpoint (CEP). This softening is indicative of an ultrasonic
dispersion phenomenon, where the anomaly in the elastic constants diminishes
with increasing measurement frequency. Fine structures were observed near the
CEP in higher frequencies. The magnetic field dependence of $C_{33}$ can be
explained by assuming a specific field dependence of the relaxation time.
Remarkably, we recorded a relaxation time of 3.5$\times$10$^{-8}$ s in the
vicinity of the CEP, which is the longest observed value among solids. These
peculiar ultrasonic properties cannot be explained solely by Ising-like
ferromagnetic fluctuations, suggesting the involvement of the quadrupole
(orbital) degree of freedom in the formation of the CEP. We discussed the
origin of these observed phenomena in relation to the magnetic ground state of
the UCoAl system.
