= Solution
Choose three representative vectors, with nonzero coordinate scale suppressed. Their <stabilizer subgroups> and <fixed-point subspaces of a group action> are
$$
\begin{array}{c|c|c|c}
\text{vector}&\text{stabilizer}&\text{order}&\text{fixed space}\\\hline
(1,0,0)&\langle r_x,\kappa_y,\kappa_z\rangle\cong D_4&8&\mathbb R(1,0,0)\\
(1,1,0)&\langle\kappa_z,\kappa_xr_z\rangle\cong C_2\times C_2&4&\mathbb R(1,1,0)\\
(1,1,1)&\langle\kappa_xr_z,\kappa_yr_x\rangle\cong S_3&6&\mathbb R(1,1,1).
\end{array}
$$
Here $\kappa_xr_z$ exchanges the first two coordinates and $\kappa_yr_x$ exchanges the last two. On the first line the <stabilizer subgroup> is every signed permutation of the two zero coordinates; fixing it forces those coordinates to vanish. On the second line it exchanges the equal coordinates and independently reflects the zero coordinate, forcing $x=y,z=0$. On the third line all coordinate permutations fix the vector, forcing $x=y=z$.
The three <subgroups> have different orders and hence cannot be conjugate. Their <group orbits> contain respectively six face-axis, twelve edge-axis and eight vertex-axis directions. Each <fixed-point subspace of a group action> is one-dimensional, so the <equivariant branching lemma> guarantees \b[three distinct symmetry types of bifurcating <equilibria>]. The <axial isotropy types of the full cube symmetry group> are precisely these: a vector with several distinct nonzero absolute coordinate values has a separate freely variable coordinate block for each such value in the <fixed-point subspace of a group action> of its <stabilizer subgroup>, giving dimension greater than one.
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