Classification of bi-Hamiltonian pairs extended by isometries

Abstract

The aim of this article is to classify pairs of first-order Hamiltonian operators of Dubrovin-Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of 2 dependent variables, and a significant new example that is an extension of a hydrodynamic type system obtained from a particular solution of the WDVV equations.