Classification of bi-Hamiltonian pairs extended by isometries

The aim of this article is to classify pairs of the 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 a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten-Dijkgraaf-Verlinde-Verlinde equations.