Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane

We discuss a non-Hamiltonian vector field  appearing in considering the partial motion of a Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In two partial cases  this vector field is expressed via Hamiltonian vector fields using a nonalgebraic deformation of the canonical  Poisson bivector on $e^*(3)$.  For the symmetric ball we also calculate variables of separation, compatible Poisson brackets, the algebra of Haantjes  operators and  $2\times2$ Lax matrices.