  A symmetry group classification for fourth-order reaction-diffusion
equations, allowing for both second-order and fourth-order diffusion terms, is
carried out. The fourth order equations are treated, firstly, as systems of
second-order equations that bear some resemblance to systems of coupled
reaction-diffusion equations with cross diffusion, secondly, as systems of a
second-order equation and two first-order equations. The paper generalizes the
results of Lie symmetry analysis derived earlier for particular cases of these
equations. Various exact solutions are constructed using Lie symmetry
reductions of the reaction-diffusion systems to ordinary differential
equations. The solutions include some unusual structures as well as the
familiar types that regularly occur in symmetry reductions, namely self-similar
solutions, decelerating and decaying traveling waves, and steady states.
