ABSTRACT
Lie's investigations were centered on problems of general integrability of differential equations by means of group theory. This is why many of his papers deal with ordinary differential equations or with linear partial differential equations of first order. In his paper on higher-order partial differential equations, Lie considered solutions invariant under groups admitted by these equations. This chapter provides an introduction to the notion of an invariant solution and the construction of sets of independent invariant solutions. It is sufficient, for purposes of illustration, to consider ordinary differential equations. The chapter considers invariant solutions with respect to a given one-parameter group or one-dimensional Lie algebra. If a differential equation admits a Lie algebra Lr of dimension r > 1, one could in principle consider invariant solutions based on one, two, etc., dimensional subalgebras of Lr. However, there are an infinite number of subalgebras, e.g., one-dimensional subalgebras.
