  Mode-coupling theory for uniformly sheared underdamped systems with an
isothermal condition is presented. As a result of the isothermal condition, it
is shown that the shear stress exhibits significant relaxation at the
alpha-relaxation regime due to the cooling effect which is accompanied by the
growth of the current fluctuation. This indicates that nonequilibrium
underdamped MCT is not equivalent to the corresponding overdamped MCT, even at
long time-scales after the early stage of the beta relaxation. It is verified
that the effect of correlations to the density-current modes is, however,
negligible in sheared thermostated underdamped systems.
