On deformations of the canonical Poisson bracket for the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems on zero-level of the area integral. I

We discuss the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems when the corresponding phase space is equivalent to cotangent bundle to dwo-dimensional sphere. In both cases Poisson bivectors are determined by $L$-tensors with non-zero torsion on the configurational space, in contrast with the well known Eisenhart–Benenti and Turiel constructions.