Integrable generalization of the associated Camassa–Holm equation

• The generalized ACH–KdV equation is integrable in the sense of Lax pair.
• The generalized ACH–KdV equation possesses the bilinear Bäcklund transformations.
• Its infinite conservation laws are constructed.
• The Darboux transformation for the equation is derived.

In this paper, we study an integrable generalization of the associated Camassa–Holm equation. The generalized system is shown to be integrable in the sense of Lax pair and the bilinear Bäcklund transformations are presented through the Bell polynomial technique. Meanwhile, its infinite conservation laws are constructed, and conserved densities and fluxes are given in explicit recursion formulas. Furthermore, a Darboux transformation for the system is derived with the help of the gauge transformation between two Lax pairs. As an application, soliton and periodic wave solutions are given through the Darboux transformation.