Abstract
The KdV-type spectral equations, which are equivalent to Z N symmetric matrix spectral equations under the Miura transformation and have the A N−1 (1) Kac-Moody algebra as a hidden symmetry, are considered. Thus the Poisson bracket, which corresponds to the adjoint of the recursion operator should be shown giving the realization of the classical extended Virasoro algebras that involve the generators of higher spins as KdV-type potentials. The Z 3 and Z 4 cases are displayed explicitly.
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