One invariant measure and different poisson brackets for two non-holonomic systems

We discuss the non-holonomic Chaplygin and the Borisov-Mamaev-Fedorov systems, for which symplectic forms are different deformations of the square root from the corresponding invariant volume form. In both cases second Poisson bivectors are determined by L-tensors with non-zero torsion on configuration space, in contrast with the well-known Eisenhart-Benenti and Turiel constructions.