On classification of integrable nonevolutionary equations

We study partial differential equations of second order (in time) that possess a hierarchy of infinitely many higher symmetries. The famous Boussinesq equation is a member of this class after the extension of the differential polynomial ring. We develop the perturbative symmetry approach in symbolic representation. Applying it, we classify the homogeneous integrable equations of fourth and sixth order (in the space derivative) equations, as well as we have found three new tenth-order integrable equations. To prove the integrability we provide the corresponding bi-Hamiltonian structures and recursion operators.