  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 is carried out,
which leads to bi-Hamiltonian structures for the same systems of equations.
Some exact solutions and Lax pairs are also constructed for the systems
considered.
