  Combining an old idea of Olver and Rosenau with the classification of second
and third order homogeneous Hamiltonian operators we classify compatible trios
of two-component homogeneous Hamiltonian operators. The trios yield pairs of
compatible bi-Hamiltonian operators whose structure is a direct generalization
of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give
rise to multi-parametric families of bi-Hamiltonian systems. We recover known
examples and we find new integrable systems whose central invariants are
non-zero; this shows that new examples are not Miura-trivial.
