  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.
