Extending Hamiltonian Operators to Get Bi-Hamiltonian Coupled KdV   Systems
  An analysis of extension of Hamiltonian operators from lower order to higher order of matrix paves a way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a consequence of bi-Hamiltonian structure, they all possess infinitely many symmetries and infinitely many conserved densities. 
Wen-Xiu Ma and Maxim Pavlov