On a two-component Camassa-Holm equation

A two-component generalization of the Camassa-Holm equation and its reduction proposed recently by Xue, Du and Geng [Appl. Math. Lett. {\bf 146} (2023) 108795] are studied. For this two-component equation, its missing bi-Hamiltonian structure is constructed and a Miura transformation is introduced so that it may be regarded as a modification of the very first two-component Camassa-Holm equation. %[Phys. Rev. E {\bf 53} (1996) ; Lett. Math. Phys. {\bf 53 } (2006)]. Using a proper reciprocal transformation, a particular reduction of this two-component equation, which admits $N-$ peakon solution, is brought to the celebrated Burgers equation.