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AKS systems and Lepage equivalent problems

The integrable systems known as "AKS systems" admit a natural formulation in terms of a Hamiltonian picture. The Lagrangian side of these systems are far less known; a version in these terms can be found in a work of Feher et al. The purpose of these notes in to provide a novel description of AKS systems in terms of a variational problem different from the usual in mechanics. Additionally, and using techniques borrowed from an article of M. Gotay, it was possible to build the Hamiltonian side of this variational problem, allowing us to establish the equivalence with the usual approach to these integrable systems.