Bilinear forms and Bäcklund transformations of the perturbation systems

A class of bilinear forms and a class of Bäcklund transformations are presented for the perturbation systems generated from perturbations around solutions of a given system of integrable equations. The stability notion of bilinear structures is introduced to guarantee their hereditariness from the original system to their perturbation systems. Two special choices of the resulting bilinear forms and Bäcklund transformations are discussed. The Korteweg-de Vries equation is chosen as a model to illustrate the general idea.