The Courant Type Algebroids, the Coadjoint Orbits, and Related Integrable Flows

Poisson structures related with the affine Courant type algebroid are analyzed, including those related with cotangent bundles on Lie group manifolds. A special attention is paid to Courant type algebroids and related R-structures on them, generated by suitably defined tensor mappings. There are constructed Lie–Poisson brackets invariant with respect to the coadjoint action of the loop diffeomorphisms group and described the related Courant type algebroids. The corresponding integrable Hamiltonian flows, generated by Casimir functionals and generalizing the so called heavenly type differential systems, describing diverse geometric structures of conformal type on finite dimensional Riemannian manifolds are described.