= Solution
For the positive-energy massless orbit choose $k=(E,E,0)$. Its connected <little group> is the one-parameter group of null rotations, isomorphic to $(\mathbb R,+)$. Its unitary irreducible representations are the characters $t\mapsto e^{i\rho t}$, $\rho\in\mathbb R$, and induction gives the massless one-particle representations. The physically usual finite-component representation has trivial little-group action, $\rho=0$; there is no $SO(2)$ helicity subgroup for a null momentum in $(2+1)$ dimensions.
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