Nonlocal symmetries of two 2-component equations of Camassa–Holm type

For a $2$-component Camassa–Holm equation, as well as a $2$-component generalization of the modified Camassa–Holm equation, nonlocal infinitesimal symmetries quadratically dependent on eigenfunctions of linear spectral problems are constructed from functional gradients of spectral parameters. With appropriate pseudopotentials, these nonlocal infinitesimal symmetries are prolonged to enlarged systems, and then explicitly integrated to generate symmetry transformations in finite form for the enlarged systems. As implementations of these finite symmetry transformations, some kinds of nontrivial solutions and Bäcklund transformations are derived for both equations.