In order to apply the Marsden–Weinstein reduction, the action of the group $G$ must be free and proper. On the other hand, if I correctly understand, the M-W reduction obtained from a given group $G$ can be used to decrease the number of degrees of freedom of a Hamiltonian $H$, provided that the Hamiltonian flow of $H$ commutes with the action of $G$.

Could you please give an example of such a Hamiltonian $H$ and of such a group $G$, whose action is not proper?

Please, try to give an example in which $G$ has the lowest possible dimension: I mean, if it is possible, provide a 1-dimensional Lie group $G$.