Bi-Hamiltonian structure of N-component Kodama equations

The author presents a simple way of constructing the second Hamiltonian operators for N-component Kodama equations. Using dimensional analysis he is led to an ansatz for the Hamiltonian operator as well as the conserved quantities in terms of ratios of polynomials. The coefficients of these polynomials are determined from the Jacobi identities. The resulting bi-Hamiltonian structure consists of generalization of Cavalcante and McKean's work (1982) for N=2 and his earlier results for N=3,4.