Conformal properties and Bäcklund transform for the associated Camassa-Holm equation

Integrable equations exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants. The most basic and important example is the KdV equation and the corresponding Schwarz-KdV equation. Other examples, including the Camassa-Holm equation and the associated Camassa-Holm equation are investigated in this Letter. It is shown that the Bäcklund transform is related to the conformal properties of these equations. Some particular solutions of the associated Camassa-Holm equation are discussed also.