  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
differential operators L=AB^{-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 Kupershmidt-Wilson theorem.
