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In this paper we revisit the classification of the gauge transformations inthe Euler top system using the generalized classical Hamiltonian dynamics ofNambu. In this framework the Euler equations of motion are bi-Hamiltonian and$SL(2, \mathbb{R})$ linear combinations of the two Hamiltonians leave theequations of motion invariant, although belonging to inequivalent Lie-Poissonstructures. Here we give the explicit form of the Hamiltonian vector fieldsassociated to the components of the angular momentum for every singleLie-Poisson structure including both the asymmetric rigid bodies and itssymmetric limits. We also give a detailed classification of the differentLie-Poisson structures recovering all the ones reported previously in theliterature.

Classification of the classical $SL(2,\mathbb{R})$ gauge transformations in the rigid body