= Solution
Let a smooth $G$-equivariant vector field have a trivial <equilibrium> branch and a critical real eigenspace $V\oplus V$, where $V$ is an <absolutely irreducible real representation> of the finite group $G$. Assume the critical <linearization> is semisimple with <eigenvalues> $\pm i\omega_0$, $\omega_0\ne0$, no other imaginary-axis <eigenvalues> are present, and the real part of the critical <eigenvalue> crosses zero transversely as the parameter varies. The <equivariant Hopf theorem> then gives, generically, a branch of small-amplitude periodic solutions for each conjugacy class of <complex axial isotropy subgroup> of $G\times S^1$ in the critical representation. The restricted <Hopf bifurcation> on its two-dimensional fixed space must be nondegenerate, with nonzero leading radial cubic coefficient. Each branch has the associated <spatiotemporal symmetry of a periodic orbit> and period tending to $2\pi/|\omega_0|$.
The spatial multiplicity means this is not the ordinary simple-<eigenvalue> Hopf hypothesis: the required critical representation is the whole two-copy isotypic component, and the semisimple pair has the corresponding symmetry multiplicity. The theorem guarantees primary branch existence and symmetry, but neither their stability nor the side of the parameter on which they exist. Those require the nonlinear coefficients and transverse perturbation analysis. A <centre manifold> and a symmetry-compatible <normal form of a dynamical system> isolate the critical dynamics and supply those coefficients.
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