  Poisson structures related with the affine Courant type algebroid are
analyzed, including \ those related with cotangent bundles on Lie group
manifolds. A special attantion 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 Rieamnnian manifolds are described.
