Lower Korteweg–de Vries equations and supersymmetric structure of the sine-Gordon and Liouville equations

A continuation of the hierarchy of the Korteweg–de Vries equation in the direction corresponding to negative powers of the spectral parameter is constructed. Among the members of this hierarchy there are equations related by a Miura transformation to the sine-Gordon and Liouville equations. The supersymmetric structure of this connection is clarified.