  We consider nonholonomic systems with symmetry possessing a certain type of
first integrals that are linear in the velocities. We develop a systematic
method for modifying the standard nonholonomic almost Poisson structure that
describes the dynamics so that these integrals become Casimir functions after
reduction. This explains a number of recent results on Hamiltonization of
nonholonomic systems, and has consequences for the study of relative equilibria
in such systems.
