  In this study we develop a systematic procedure to construct a Poisson
operator that describes the dynamics of a three dimensional nonholonomic
system. Instead of reducing by symmetry the antisymmetric operator that links
the energy gradient to the velocity on the tangent bundle, the system is
embedded in a larger space. Here, the extended antisymmetric operator, which
preserves the original equations of motion, satisfies the Jacobi identity in a
conformal fashion. Thus, a Poisson operator can be obtained by a further time
reparametrization. Such Poissonization does not rely on the specific form of
the Hamiltonian function. The theory is applied to calculate the equilibrium
distribution function of a non-Hamiltonian ensemble.
