Bi-Hamiltonian structures of WDVV-type

We study a class of nonlinear PDEs that admit the same bi-Hamiltonian structure as WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle--Potemin 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 parameterising the first-order Hamiltonian operator.