Hierarchy of Hamilton equations on Banach Lie–Poisson spaces related to restricted Grassmannian

We consider the Banach Lie–Poisson space and its complexification , where the first one of them contains the restricted Grassmannian Grres as a symplectic leaf. Using the Magri method we define an involutive family of Hamiltonians on these Banach Lie–Poisson spaces. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases we give the explicit solutions.