 Equivariant branching lemma

For an absolutely irreducible real symmetry representation, an equivariant steady-state bifurcation with a simple transverse crossing of the scalar linear eigenvalue generically has an equilibrium branch for each axial isotropy type. Restricting the vector field to its one-dimensional fixed-point subspace reduces the existence problem to a scalar bifurcation equation. The lemma guarantees these branches; it does not exclude branches with smaller isotropy or higher-dimensional fixed-point subspaces.