On novel Hamiltonian description of the nonholonomic Suslov problem

The paper constructs the Suslov vector field X, invariants f1,f2, the Darboux polynomial D, and three invariant Poisson bivectors P1,P2,P3; it then exhibits two rank-4 invariant Poisson structures Pa and Pb and shows X is Hamiltonian with respect to each, with Padf1 = 2c5 f1 X, Padf2 = 0 and Pbdf1 = 2c5 f1 X, Pbdf2 = 4c5 f2 X (equations (2.8),(2.9) and ensuing identities) . The candidate solution verifies the Jacobi and invariance conditions directly in coordinates and reproduces these action identities, then deduces the Hamiltonians and Casimirs. The only discrepancy is a sign in the paper’s printed Casimir for Pb: the paper states Cb = ln f1^2 + ln f2, but with the given Pbdf1 and Pbdf2 this yields 8c5 X rather than 0; the candidate correctly identifies and fixes this to Cb = ln f2 − 2 ln f1 . Aside from this typographical sign error, the arguments and conclusions agree, including the rank-four property and the existence of Darboux coordinates on leaves via the constant-rank theorem of Lie .