  The Moyal-Lax representation and the Moyal momentum algebra are introduced
and 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.
