Two-component generalizations of the Camassa-Holm equation

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.