THE HAMILTONIAN STRUCTURES ASSOCIATED WITH A GENERALIZED LAX OPERATOR

It is shown that with every Lax operator, which is a pseudodifferential operator of nonzero leading order, is associated a KP hierarchy. For each such operator, we construct the second Gelfand-Dikii bracket associated with the Lax equation and show that it defines a Hamiltonian structure. When the leading order is positive the corresponding compatible first Hamiltonian structure, which turns out, in general, to be different from the naive first Gelfand-Dikii bracket is derived. The corresponding Hamiltonian structures for the constrained Lax operator, where the next to leading-order term vanishes or has a constant coefficient, is discussed.