Simplifying the Form of Lie Groups Admitted by a Given Differential Equation

Rather general results are obtained for determining the nature of infinitesimal generators which can be admitted by a given differential equation. Simple criteria are given to determine whether or not (1) the infinitesimals of independent variables can depend only on independent variables and (2) the infinitesimal of the dependent variable can depend at most linearly on the dependent variable. Many examples are given. A trivial consequence of this paper is that an admitted Lie group, for any linear partial differential equation of at least second order, must have infinitesimals of independent variables depending only on independent variables and the infinitesimal of the dependent variable depends at most linearly on the dependent variable. This latter result previously has only been proved for a second order linear PDE. #~‘I 1990 Academic Press, Inc