On novel Hamiltonian description of the nonholonomic Suslov problem

We present some new Poisson bivectors that are invariants by the flow of the nonholonomic Suslov problem. Two rank four invariant Poisson bivectors have globally defined Casimir functions and, therefore, define cubic Poisson brackets on the five dimensional state space with standard symplectic leaves. For the Suslov gyrostat in the potential field we found rank two Poisson bivectors having only two globally defined Casimir functions and, therefore, we say about formal Hamiltonian description in these cases.