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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.