Lie group symmetries as integral transforms of fundamental solutions

We obtain fundamental solutions for PDEs of the form ut=σxγuxx+f(x)ux−μxru by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.