KP hierarchy and Calogero–Moser hierarchy

We consider trigonometric solutions of the KP hierarchy. It is known that their poles move as particles of the Calogero–Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution of the poles with respect to the $k$-th hierarchical time of the KP hierarchy is governed by a Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the trigonometric Calogero–Moser hierarchy.

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