Rotational Lie-Poisson structure on R3 ($\{x,y\}=z$)

The rotational Lie-Poisson bracket on ${\mathbb{R}}^3$ isIts Casimir is $x_1^2+x_2^2+x_3^2$, and every sphere centered at the origin is a two-dimensional symplectic leaf on which $SO(3)$ acts symplectically.