We present a detailed discussion of the infinit esimal symmetries of the Hamilton-Jacobi equation (an arbitrary first order partial equation) Our presentation clucidates the role played by the characteristic system in determining the symmetries. We then specialize to the case of a free particle in one space and one time dimension, and study of local Lie group of point transformations locally isomorphie to O(3,2). We show that the separation of variables of the corresponding Hamilton-Jacobi equation n the form of a sum is related to orbits in the Schrödinger subalgebra of c(3,2). The remaining orbits of o(3,2) yield symmetry related solutions which separate in more complicated product forms. Finally some connections with the primordial equation of hydrodynamics (without force terms) are made.