Quasi-probability distributions for the simplest dynamical groups

We prove that the Wigner-Stratonovich-Agarwal operator that defines the quasi-probability distribution on the sphere [for the SU(2) dynamical group] can be written as an integral of the SU(2) (irreducible unitary) representation element with respect to a single variable that labels the orbits in the coadjoint representation. This allows us to consider contractions of the SU(2) quasi-probability distribution to the cases of the Heisenberg-Weyl group and the two-dimensional Euclidean group.