  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.
