Bäcklund transformations of solutions of nonlinear evolution equations and the Lie Bianchi transformation

Using the Maurer Cartan structural equations of the Lie group SL(2, ) it is shown that the auto-Bäcklund transformations of solutions of the Sine Gordon, the Korteweg—de Vries and the modified Korteweg—de Vries equation can be derived from the same Lie Bianchi transformation sending surfaces of constant negative curvature into surfaces of the same constant negative curvature.