The Marsden-Weinstein-Meyer theorem is expressed in a too general way to be understood by a mean square physicist, as me. However, if we limit the scope to a Hamiltonian mechanics, it should be possible to express it in the Hamiltonian language, at least to some extent.

So my question is what the M.W.M. theorem says on the following problem. I have a Hamiltonian $H$, and I know that it is in involution with another function $J$, i.e. $\{H,J\}=0$. 

1) What are the additional hypothesis that must be met by $H$ and $J$, so that the M.W.M. theorem can be applied?

2) Under those hypothesis, what is the thesis?

But: since the problem is expressed in simple and practical terms, also the answer must be absolutely practical. Derivatives, Poisson parenthesis, solutions of differential equations are allowed; everything that can be followed by a physicists is allowed; but no Lie group, no symplectomorphism, no coadjoint action is allowed in the answer. Where this is not possible, examples should be given.