Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

SYSTEMS OF HESS–APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY

We start with a review of a class of systems with invariant relations, so called systems of Hess–Appel'rot type that generalizes the classical Hess–Appel'rot rigid body case. The systems of Hess–Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess–Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the Zhukovskii property: These are Hamiltonian systems on a symplectic manifold M with actions of two groups G and K; the systems are assumed to be K-invariant and to have invariant relation Φ = 0 given by the momentum mapping of the G-action, admitting two types of reductions, a reduction to the Poisson manifold P = M/K and a partial reduction to the symplectic manifold N0 = Φ-1(0)/G; final and crucial assumption is that the partially reduced system to N0 is completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess–Appel'rot type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess–Appel'rot type — the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian Gr+(n, 2) has a natural interpretation within Zhukovskii property and that it is equivalent to a partial reduction of certain system of Hess–Appel'rot type. We perform a classical and algebro-geometric integration of the system in dimension four, as an example of a known isoholomorphic system — the Lagrange bitop. The paper presents a lot of examples of systems of Hess–Appel'rot type, giving an additional argument in favor of further study of this class of systems.

COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.