Tri-Hamiltonian duality between the Wadati-Konno-Ichikawa hierarchy and the Song-Qu-Qiao hierarchy.

A linear eigenvalue problem is constructed for the gradients of the eigenvalue of the Wadati-Konno-Ichikawa (WKI) spectral problem. This in turn leads to a compatible pair of Hamiltonian operators, which endows the WKI hierarchy with a bi-Hamiltonian structure. Furthermore, a proper recombination of these two Hamiltonian operators, or the duality of this bi-Hamiltonian structure, generates a coupled system of Camassa-Holm type, which is nothing but a system proposed by Song, Qu, and Qiao [J. Math. Phys. 52, 013503 (2011)] recently. [ABSTRACT FROM AUTHOR]