In this paper, we mainly study the geometric background, integrability and peaked solutions of a $(1+n)$-component Camassa–Holm (CH) system and some related multi-component integrable systems. Firstly, we show this system arises from the invariant curve flows in the Möbius geometry and serves as the dual integrable counterpart of a geometrical $(1+n)$-component Korteweg–de Vries system in the sense of tri-Hamiltonian duality. Moreover, we obtain an integrable two-component modified CH system using a generalized Miura transformation. Finally, we provide a necessary condition, under which the dual integrable systems can inherit the Bäcklund correspondence from the original ones.

在本文中，我们主要研究了一个几何背景、可积性和峰值解。 $(1+n)$-分量 Camassa–Holm (CH) 系统和一些相关的多分量可积系统。首先，我们展示了这个系统是由莫比乌斯几何中的不变曲线流产生的，并且作为几何的对偶可积对应物$(1+n)$-分量 Korteweg-de Vries 系统在三汉密尔顿对偶的意义上。此外，我们使用广义 Miura 变换获得了一个可积的双组分改进 CH 系统。最后，我们提供了一个必要条件，在该条件下，对偶可积系统可以继承原始系统的 Bäcklund 对应关系。