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Poisson structures related with the affine Courant type algebroid areanalyzed, including \ those related with cotangent bundles on Lie groupmanifolds. A special attantion is paid to Courant type algebroids and relatedR-structures \ on them, generated by suitably defined tensor mappings. \ Thereare constructed Lie-Poisson brackets invariant with respect to the coadjointaction of the loop diffeomorphisms group and described the related Courant typealgebroids. \ The corresponding integrable Hamiltonian flows, generated byCasimir functionals and generalizing the so called heavenly type differentialsystems, describing diverse geometric structures of conformal type on finitedimensional Rieamnnian manifolds are described.

The Courant type algebroids, the coadjoint orbits and related integrable flows