  We construct 1-parameter families of well-known solutions to the Yamabe
problem defined on Aloff-Wallach Spaces to determine bifurcation instants for
these homogeneous spaces by examining changes in the Morse index of these
metrics as the parameter varies over the positive real numbers. A bifurcation
point for such families is an accumulation point of other solutions to the
Yamabe problem, while a local rigidity point is an isolated solution of this
problem, i.e., it is not a bifurcation point.
