Canonical property of the Miura maps between the MKP and KP hierarchies,continuous and discrete

Apart from the case of the KP hierarchy, all known Miura maps between integrable Hamiltonian systems had been proven to be canonical. The remaining KP case is settled below. As a corollary, it is shown that the KP hierarchy is a factor — hierarchy of the mKP one, with the kernel consisting of a single scalar field. A discrete mKP hierarchy and the associated Miura map are constructed, and the latter is shown to be canonical as well. As in the continuous case, this implies that one can extend the discrete KP hierarchy by a single new field into an extended discrete KP hierarchy in such a way that the extended discrete Miura map mKPeKP is a canonical isomorphism.