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.
Hamiltonian Dynamics on Lie Algebroids, Unimodularity and Preservation of Volumes
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In this paper we discuss the relation between the unimodularity of a Lie algebroid $\tau_{A}: A\to Q$ and the existence of invariant volume forms for the hamiltonian dynamics on the dual bundle $A^{*}$. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.
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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.