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Hidden symmetries in a covariant Hamiltonian formulation are investigatedinvolving gauge covariant equations of motion. The special role of theStackel-Killing tensors is pointed out. A reduction procedure is used to reducethe original phase space to another one in which the symmetries are dividedout. The reverse of the reduction procedure is done by stages performing theunfolding of the gauge transformation followed by the Eisenhart lift inconnection with scalar potentials.

Hidden symmetries in a gauge covariant approach, Hamiltonian reduction and oxidation