Bäcklund transformation of partial differential equations from the Painlevé–Gambier classification. I. Kaup–Kupershmidt equation

Among the 50 nonlinear second-order differential equations of Painlevé and Gambier, those which are linearizable provide a natural scheme for deriving the Lax pair and the Darboux transformation of a nonlinear partial differential equation when the order of the scattering problem is three. This new method allows one to obtain, from singularity analysis only, the Bäcklund transformation of the Kaup–Kupershmidt equation, a result which was missing until now.