Gardner's deformations of the Boussinesq equations

Using the algebraic method of Gardner's deformations for completely integrable systems, we construct the recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these systems, we obtain new integrable equations adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.