Lie-Backlund groups and the linearisation of differential equations

It is shown that groups of Lie-Backlund (LB) transformations which depend on non-local variables are related by a change of variables to the LB tangent transformations of Ibragimov and Anderson (1979), involving no more than arbitrary-order derivatives. The transformation of any LB symmetry operator by an invertible change of variables is discussed. It is pointed out that once a differential equation admits an LB operator, then a large number of 'secondary' equations will admit the same operator. The LB theory involving non-local variables and the notion of secondary equations are used to characterise group theoretically the linearisation of the Burgers equation, ut+uux-uxx=0, and of the ODE uxx+ omega 2(x)u+Ku-3=0.