We analyse the underlying nonlinear partial differential equation which arises in the study of the gravitating flat fluid plates of embedding class one. Our interest in this equation lies in discussing new solutions that can be found by means of Lie point symmetries. The method utilized reduces the partial differential equation to an ordinary differential equation according to the Lie symmetry admitted. We show that a class of solutions found previously can be characterized by a particular Lie generator. Several new families of solutions are found explicitly. In particular, we find the relevant ordinary differential equation for all one-dimensional optimal subgroups; in several cases the ordinary differential equation can be solved in general. We are in a position to characterize particular solutions with a linear barotropic equation of state.