On embedding the $1:1:2$ resonance space in a Poisson manifold

The Hamiltonian actions of $\S ^{1}$ on the symplectic manifold $\mathbb {R}^{6}$ in the $1:1:-2$ and $1:1:2$ resonances are studied. Associated to each action is a Hilbert basis of polynomials defining an embedding of the orbit space into a Euclidean space $V$ and of the reduced orbit space $J^{-1}(0)/\S ^{1}$ into a hyperplane $V_{J}$ of $V$, where $J$ is the quadratic momentum map for the action. The orbit space and the reduced orbit space are singular Poisson spaces with smooth structures determined by the invariant functions. It is shown that the Poisson structure on the orbit space, for both the $1:1:2$ and the $1:1:-2$ resonance, cannot be extended to $V$, and that the Poisson structure on the reduced orbit space $J^{-1}(0)/\S ^{1}$ for the $1:1:-2$ resonance cannot be extended to the hyperplane $V_{J}$.