  It is well known that the centerless W_{1+\infty} algebra provides a
hamiltonian structure for the KP hierarchy. In this letter we address the
question whether the centerful version plays a similar r\^ole in any related
integrable system. We find that, surprisingly enough, the centrally extended
W_{1+\infty} algebra yields yet another Poisson structure for the same standard
KP hierarchy. This is proven by explicit construction of the infinitely many
new hamiltonians in closed form.
