= Solution
For $P^2=m^2>0$, choose the future-pointing rest momentum $p^\mu=(m,0,0,0)$. Its <little group> is the rotation group $SO(3)$, or $SU(2)$ on the double cover. Its finite-dimensional unitary irreducible representations are labelled by
$$
j=0,\frac12,1,\frac32,\ldots
$$
and have dimension $2j+1$. These are the spin states of a massive particle.
For $P^2=0$, choose $p^\mu=(E,0,0,E)$. Its little group is the two-dimensional Euclidean group
$$
ISO(2)=SO(2)\ltimes\mathbb R^2.
$$
Nontrivial unitary action of the translation subgroup produces infinite-dimensional <continuous-spin representation>[continuous-spin representations]. For the finite-helicity particles observed in ordinary relativistic field theory, that subgroup acts trivially, leaving a one-dimensional representation of rotations about $\mathbf p$. Such representations are labelled by helicity.
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