Properties of Moyal–Lax representation

The properties of standard and the nonstandard Moyal–Lax representations are systematically investigated. It is shown that the Moyal–Lax equation can be interpreted as a Hamiltonian equation and can be derived from an action. We show that the parameter of non-commutativity, in this case, is related to the central charge of the second Hamiltonian structure of the system. The Moyal–Lax description leads in a natural manner to the dispersionless limit and provides the second Hamiltonian structure of dispersionless integrable models, which has been an open question for sometime.