 Solution

In a Hopf critical eigenspace, equip the real space $V\oplus V$ with the complex structure defined by the temporal phase action. A subgroup $\Sigma \subseteq G\times S^1$ is a complex axial isotropy subgroup when it is the full stabilizer of a nonzero vector andThe isotropy is for the combined spatial and temporal action, so its elements encode spatiotemporal symmetry of a periodic orbit, not necessarily pointwise spatial symmetry. “Complex dimension” here refers to the Hopf complex structure; it need not agree with independently chosen spatial complex coordinates. Any kernel of the combined action belongs to every actual isotropy subgroup.