Multiple Darboux–Bäcklund transformations via truncated Painlevé expansion and Lie point symmetry approach

For a given truncated Painlevé expansion of an arbitrary nonlinear Painlevé integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, which is proved to be localized to Lie point symmetries for suitable prolonged systems. Taking the Korteweg–de Vries equation as an example, the n-th binary Darboux–Bäcklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.