  Borisov, Mamaev and Kilin have recently found certain Poisson structures with
respect to which the reduced and rescaled systems of certain non-holonomic
problems, involving rolling bodies without slipping, become Hamiltonian, the
Hamiltonian function being the reduced energy. We study further the algebraic
origin of these Poisson structures, showing that they are of rank two and
therefore the mentioned rescaling is not necessary. We show that they are
determined, up to a non-vanishing factor function, by the existence of a system
of first-order differential equations providing two integrals of motion. We
generalize the form of that Poisson structures and extend their domain of
definition. We apply the theory to the rolling disk, the Routh's sphere, the
ball rolling on a surface of revolution, and its special case of a ball rolling
inside a cylinder.
