= Axial subgroup
For a steady-state <equivariant dynamical system>, an axial <subgroup> is a <stabilizer subgroup> whose real <fixed-point subspace of a group action> has <dimension> one. Requiring an actual <stabilizer subgroup> matters: a smaller <subgroup> can have the same fixed line without being the full symmetry of any point on that line. The fixed line reduces the steady-state bifurcation equation to one real amplitude.
In an equivariant <Hopf bifurcation>, a <complex axial isotropy subgroup> is instead a <stabilizer subgroup> for the spatial <symmetry group> combined with the common phase <circle action>, with one-dimensional complex <fixed-point subspace of a group action>. Distinct <conjugacy classes> of such <subgroups> can describe distinct types of bifurcating branches.
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