On co-adjoint Lie groupoids of product form, Hamiltonian integrability is claimed to reduce to the integrability of a Hamiltonian system on a co-adjoint orbit, but the derivation is not sound.
Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics, vol
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A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids
On co-adjoint Lie groupoids of product form, Hamiltonian integrability is claimed to reduce to the integrability of a Hamiltonian system on a co-adjoint orbit, but the derivation is not sound.