ABSTRACT
This chapter contains a survey on Liouville correspondences between integrable hierarchies. A Liouville transformation between the corresponding isospectral problems induces a Liouville correspondence between their flows and Hamiltonian functionals. As prototypical examples, we construct Liouville correspondences for the modified Camassa-Holm, the Novikov, the Degasperis-Procesi, the two-component Camassa-Holm and the two-component Novikov (Geng-Xue) hierarchies. In addition, a new Liouville correspondence for a certain dual Schrödinger integrable hierarchy is presented.
