= Axial isotropy of rolls and hexagons
Write the translation <group action> as $(A,B,C)\mapsto(e^{i\theta_1}A,e^{i\theta_2}B,e^{-i(\theta_1+\theta_2)}C)$. Modulo lattice translations, the <stabilizer subgroup> of $(h,h,h)$, with nonzero real $h$, is the sixfold rotation group. Its <fixed-point subspace of a group action> is the real line $(a,a,a)$. The <stabilizer subgroup> of $(r,0,0)$, with nonzero real $r$, consists of the circle $\theta_1=0$ together with a half-turn; its <fixed-point subspace of a group action> is the real line $(a,0,0)$. The <equivariant branching lemma> therefore supplies generic branches of both axial types.
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