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In the case of three primary fields, the associativity equations (the WDVV equations of the two-dimensional topological quantum field theory) can be represented as integrable nondiagonalizable systems of hydrodynamic type (O.I. Mokhov). There arose the classification problem of the existence of a local homogeneous first-order Hamiltonian structure of the Dubrovin-Novikov type for systems of hydrodynamic type which are equivalent to the associativity equations. O.I Mokhov and the author have completely solved this problem. These results will be presented.