Abstract
Conditions necessary for pairs of Hamiltonian operators of first through fifth order to form Hamiltonian pairs are computed. In the case of bi-Hamiltonian systems involving a Hamiltonian pair, one operator of which is a first-, third-, or a certain type of fifth-order operator, a technical condition in Magri's Theorem can be dropped. © 1991 American Institute of Physics. © 2017 Elsevier B.V., All rights reserved.