Lie Subalgebras, Reduction of Order, and Group-Invariant Solutions

One of the great utilities of Lie symmetries of differential equations is in their use to reduce the order of ordinary differential equations and partial differential equations to ordinary differential equations. This process is guided by the Lie algebra of the symmetries admitted by the equation under study. It is important in these reductions to use appropriate symmetries. This will ensure that a sufficient number remain after the initial reduction to allow subsequent reductions. By considering two- and three-dimensional subalgebras of the Lie algebra of the symmetries of differential equations, we indicate the fate of these symmetries under various reductions. In doing so, it is clear as to which symmetries need be prioritized in reduction.