  We develop the reducing multiplier theory for a special class of nonholonomic
dynamical systems and show that the non-linear Poisson brackets naturally
obtained in the framework of this approach are all isomorphic to the
Lie-Poisson $e(3)$-bracket. As two model examples, we consider the Chaplygin
ball problem on the plane and the Veselova system. In particular, we obtain an
integrable gyrostatic generalisation of the Veselova system.
