In this work, we apply Lie Group Theory (LGT) to aid in solving and understanding equations arising from the Forward Master Equation formulation for the neutron number distribution in a zero-dimensional setting. In particular, we focus our LGT study on a first-order hyperbolic partial differential equation satisfied by the probability generating function. We show the connection between solutions to the symmetry determining equations with established analytical solutions given by Bell and by Prinja and Souto. We derive global transformations for isolated neutron fission chains as well as neutron sources and provide a physical interpretation of the results throughout.