Classical r-matrices and the four-wave interaction equation

A classical r-matrix in conjunction with a zero-curvature equation is used to generate a new nonlinear integrable system known as a four-wave interaction equation. On reduction it yields a new integrable Klein-Gordon system. These equations turn out to be bi-Hamiltonian. Finally, it is indicated how the modified equations can be obtained by the zero-curvature equation. Our system is very similar to those studied by Tsuchida and Wadati.