Matrix Kadomtsev–Petviashvili Hierarchy and Spin Generalization of Trigonometric Calogero–Moser Hierarchy

We consider solutions of the matrix Kadomtsev–Petviashvili (KP) hierarchy that are trigonometric functions of the first hierarchical time $t_1=x$ and establish the correspondence with the spin generalization of the trigonometric Calogero–Moser system at the level of hierarchies. Namely, the evolution of poles $x_i$ and matrix residues at the poles $a_i^\alpha b_i^\beta $ of the solutions with respect to the $k$th hierarchical time of the matrix KP hierarchy is shown to be given by the Hamiltonian flow with the Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the spin trigonometric Calogero–Moser system with coordinates $x_i$ and with spin degrees of freedom $a_i^\alpha $ and $b_i^\beta $. By considering the evolution of poles according to the discrete time matrix KP hierarchy, we also introduce the integrable discrete time version of the trigonometric spin Calogero–Moser system.