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Nonholonomic systems are described by the Lagrange-d’Alembert principle. The presence of symmetry leads to a reduced d’Alembert principle and to the Lagrange-d’Alembert-Poincaré equations. First, we briefly recall from previous works how to obtain these reduced equations for the case of a thick disk rolling on a rough surface using a three-dimensional abelian group of symmetries. The main results...