Symplectic reduction of an isotropic oscillator ($H^{-1}(E)/S^1\cong {\mathbb{CP}}^{n-1}$)

For $H={\sum }_j(q_j^2+p_j^2)/2$ and $E>0$, the oscillator flow is a free circle action on the sphere of radius $\sqrt{2E}$. Its quotient is Complex projective space. For two modes, the reduced form in $w=z_2/z_1$ is $iEdw\wedge d\bar{w}/(1+|w{|}^2{)}^2$, of total area $2\pi E$. Reduction at zero energy is singular instead of a smooth quotient of the asserted dimension.