Bi-Hamiltonian structures of WDVV-type

We study a class of nonlinear partial differential equations (PDEs) that admit the same bi-Hamiltonian structure as the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle–Potëmin form, which are compatible. Using various equivalence groups, we classify such equations in two-component and three-component cases. In a four-component case we add further evidence to the conjecture that there exists only one integrable system of the above type. Finally, we give an example of the six-component system with required bi-Hamiltonian structure. To streamline the symbolic computation, we develop an algorithm to find the aforementioned Hamiltonian operators, which includes putting forward a conjecture on the structure of the metric parameterizing the first-order Hamiltonian operator.