  We continue the study of integrability of bi-Hamiltonian systems with a
compatible pair of local Poisson structures (H_0,H_1), where H_0 is a strongly
skew-adjoint operator. This is applied to the construction of some new two
field integrable systems of PDE by taking the pair (H_0,H_1) in the family of
compatible Poisson structures that arose in the study of cohomology of moduli
spaces of curves.
