  In this paper, we study explicit correspondences between the integrable
Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and
Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations
between the isospectral problems of the Novikov and Sawada-Kotera equations,
and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt
equations relate the corresponding hierarchies, in both positive and negative
directions, as well as their associated conservation laws. Combining these
results with the Miura transformation relating the Sawada-Kotera and
Kaup-Kupershmidt equations, we further construct an implicit relationship which
associates the Novikov and Degasperis-Procesi equations.
