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.