= Solution
Let a finite or compact group $G$ act linearly on a real space $V$, and let $\dot x=F(x,\mu)$ be a smooth <equivariant dynamical system>: $F(gx,\mu)=gF(x,\mu)$. Assume $F(0,\mu)=0$. The representation is an <absolutely irreducible real representation> when its complexification is irreducible; equivalently, every real linear operator commuting with $G$ is a scalar multiple of the identity. Consequently the <linearization> at the trivial branch is $D_xF(0,\mu)=\alpha(\mu)I$. A generic steady-state crossing has $\alpha(0)=0$ and $\alpha'(0)\ne0$, with stable noncritical directions removed by a <centre manifold> if this is a reduction of a larger system.
The <isotropy group> of a vector $x$ is $\Sigma_x=\{g\in G:gx=x\}$. For a subgroup $\Sigma$, its <fixed-point subspace of a group action> is $\operatorname{Fix}(\Sigma)=\{x:\sigma x=x\text{ for every }\sigma\in\Sigma\}$. Its <normaliser> is
$$
N_G(\Sigma)=\{g\in G:g\Sigma g^{-1}=\Sigma\}.
$$
The <normaliser> preserves $\operatorname{Fix}(\Sigma)$, and $N_G(\Sigma)/\Sigma$ describes its residual symmetry. A steady-state <axial subgroup> is an actual isotropy subgroup with $\dim_{\mathbb R}\operatorname{Fix}(\Sigma)=1$.
The <equivariant branching lemma> states that, under the smoothness, equivariance, absolute irreducibility, and transverse scalar-<eigenvalue> crossing hypotheses above, each conjugacy class of axial isotropy subgroup gives a branch of nonzero <equilibria> in its fixed-point line through $(0,0)$. The generic leading nonlinear coefficient determines which parameter side the branch occupies and whether it is transcritical-like or pitchfork-like; the theorem does not assert that these branches are stable or exhaust every possible branch. The usual nondegeneracy condition excludes a vanishing leading nonlinear coefficient. Conjugate subgroups produce symmetry-related copies of a branch, rather than new isotropy types.
To explain the dynamical role of the <fixed-point subspaces of a group action>, if $x$ is fixed by $\Sigma$, then $\sigma F(x,\mu)=F(\sigma x,\mu)=F(x,\mu)$ for every $\sigma\in\Sigma$. Thus the vector field is tangent to that subspace, and uniqueness makes it flow invariant. On a fixed-point line one solves a scalar <equilibrium> equation and determines radial stability. Stability in the full space additionally requires all transverse <eigenvalues> to be stable. Higher-dimensional fixed-point spaces support their own restricted dynamics and may contain symmetry-constrained connecting orbits.
Finally, the <normalizer action determines axial branch parity>: if $N_G(\Sigma)/\Sigma$ contains an element acting as $x\mapsto-x$ on the fixed line, its scalar vector field must be odd, so a generic cubic term gives a <pitchfork bifurcation>. If that residual sign reversal is absent, a quadratic term is permitted and can give a transcritical-like branch. This is why fixed-space dimension alone does not determine either the branch type or its full stability.
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