Lemma: Spinless fixed set

For $0<\mu <2\pi$, $J_{\mu }$ fixes precisely the transverse time axis, hence its product lift fixes the worldsheet pointwise. The conjugate $H_A(\mu )$ fixes ${\mathcal{W}}_A$ pointwise and preserves the transported rest vector in its linear part.

Proof. A nontrivial planar rotation has only the origin as fixed spatial point. Time and axial coordinates are unchanged. Conjugation transports the fixed set and rest vector.