The constrained KP hierarchy and the generalised Miura transformation

Recently much attention has been paid to the restriction of KP to the submanifold of operators which can be represented as a ratio of two purely differetial operators L = ABf-1. Whereas most of the aspects concerning this reduced hierarchy, like the Lax flows and the Hamiltonians, are by now well understood, there still lacks a clear and conclusive statement about the associated Poisson structure. We fill this gap by placing the problem in a more general framework and then showing how the required result follows from an interesting property of the second Gelfand-Dickey brackets under multiplication and inversion of Lax operators. As a byproduct we give an elegant and simple proof of the generalised Kuperschmidt-Wilson theorem.