Can we always find a Quasi-Probability distribution representation for density operator of any physical system? For a pure, quantum optical system, the QPD formalism is straightforward and possible because of the existence of coherent states. 
The question is whether we can find coherent states for any quantum mechanical system, not necessarily quantum optical system, and hence find its QPD representation.
 A: There is a famous result by Onofri: 


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*Onofri, Enrico. "A note on coherent state representations of Lie groups." Journal of Mathematical Physics 16.5 (1975): 1087-1089.


which shows (in a nutshell) that if you have a representation of a group and this representation has a lowest (or highest) weight state then it's possible to define coherent states as those states in the orbit of the lowest weight state; Onofri observes that from this you get a  (classical) phase space for free.
If the group is not a semi-simple Lie group (v.g. $E(2)$), then the definiton a coherent state is not so clear, but in can in some cases be done: the obvious example is $HW(1)$, i.e. for the harmonic oscillator (there’s obviously a “lowest weight state” there: the ground state $\vert 0\rangle$.)
So if you have coherent states you certainly have $Q$-functions, and if you have $Q$-functions there should exists a (self-dual) Wigner function.  However, there are some serious technical difficulties, especially for non-compact groups.  Even for compact group the effort is substantial, as you can sense from Sec. 7 of 


*Klimov, Andrei B., José Luis Romero, and Hubert de Guise. "Generalized SU(2) covariant Wigner functions and some of their applications." Journal of Physics A: Mathematical and Theoretical 50.32 (2017): 323001, which is open-access (maybe paywalled after all...)


There's quite a bit of literature on generalized coherent states: a good example is 


*Gilmore, Robert. "Coherent states for bosons and fermions: A tutorial." Progress in Particle and Nuclear Physics 9 (1983): 479-494 (unfortunately behind a paywall).


See also Zhang, Wei-Min, and Robert Gilmore. "Coherent states: theory and some applications." Reviews of Modern Physics 62.4 (1990): 867. if you can access it.
