Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal

By the application of the coinduction method as well as the Magri method to the ideal of real Hilbert–Schmidt operators we construct the hierarchies of integrable Hamiltonian systems on Banach Lie–Poisson spaces which consist of such types of operators. We also discuss their algebraic and analytic properties and solve them in dimensions, N=2,3,4N=2,3,4.