Nonlocal Symmetries and Geometric Integrability of Multi-Component Camassa–Holm and Hunter–Saxton Systems

We present the multi-component Hunter–Saxton and μ−Camassa–Holm systems. It is shown that the multi-component Camassa–Holm, Hunter–Saxton and μ-Camassa–Holm systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws can be directly constructed. For the three-component Camassa–Holm and Hunter–Saxton systems, their nonlocal symmetries depending on the pseudo-potentials are obtained.