  We analyse the underlying nonlinear partial differential equation which
arises in the study of 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 utilised 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
characterised 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 characterise particular solutions with a linear barotropic equation of
state.

General Relativity and Quantum Cosmology