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.