= Solution
For a massive positive-energy orbit, choose $p_*=(m,0,0)$ with $m>0$. Its <little group> is $SO(2)$, whose <unitary irreducible representation>[unitary irreducible representations] are the characters $e^{is\theta}$. Induction gives one-particle states $|p,s\rangle$ on the mass shell $p^2=m^2$, $p^0>0$, labeled by mass and spin. For the Poincare group itself, single-valuedness gives $s\in\mathbb Z$; its double cover permits half-integers, and its universal cover permits any real $s$, producing <anyon>[anyonic spin] in $(2+1)$ dimensions.
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