  We study Lifshitz tails for random Schr\"odinger operators where the random
potential is alloy type in the sense that the single site potentials are
independent, identically distributed, but they may have various function forms.
We suppose the single site potentials are distributed in a finite set of
functions, and we show that under suitable symmetry conditions, they have
Lifshitz tail at the bottom of the spectrum except for special cases. When the
single site potential is symmetric with respect to all the axes, we give a
necessary and sufficient condition for the existence of Lifshitz tails. As an
application, we show that certain random displacement models have Lifshitz
singularity at the bottom of the spectrum, and also complete the study of
continuous Anderson type models undertaken in arXiv : 0804.4079
