  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.
