Compatibility conditions between Dubrovin-Novikov integrability operators and systems of PDEs

Compatibility conditions between Hamiltonian, symplectic and recursion operators and a given system of PDEs can be systematically derived by a method that was introduced by Kersten, Krasil'shchik and Verbovetsky in 2003. The method is geometrically invariant, and when applied to homogeneous symplectic or Hamiltonian operators of Dubrovin-Novikov type, it produces geometric conditions of compatibility between systems of PDEs and corresponding operators. We recover old results like Tsarev's compatibility conditions between a hydrodynamic-type system and a first-order local Dubrovin-Novikov Hamiltonian operator, and we find new results, especially in (but not limited to) the case of third-order Hamiltonian operators.
Joint work with E.V. Ferapontov, M.V. Pavlov.