= Solution
Coordinate <reflection> in $x_i$ requires the $i$th component to be odd in $x_i$, and <reflections> in the other coordinates require it to be even in those coordinates. Thus no constant or quadratic terms are permitted. The only cubic monomials in component $i$ are $x_i^3$ and $x_ix_j^2$ for $j\ne i$. Coordinate permutations equate their coefficients in all components. After using the transverse linear <eigenvalue> as parameter, the <normal form> is
$$
\boxed{\begin{aligned}
\dot x&=\mu x+a x^3+b x(y^2+z^2)+\cdots,\\
\dot y&=\mu y+a y^3+b y(z^2+x^2)+\cdots,\\
\dot z&=\mu z+a z^3+b z(x^2+y^2)+\cdots.
\end{aligned}}
$$
The remainder starts at fifth spatial order, together with parameter-dependent corrections such as $\mu|x|^3$. This is the <cubic equivariants of the full cube symmetry group> form. On a branch with $m=1,2,3$ equal nonzero absolute coordinates $s$, respectively,
$$
\mu+[a+(m-1)b]s^2=0.
$$
Hence the three leading branch amplitudes are $s^2=-\mu/a$, $-\mu/(a+b)$ and $-\mu/(a+2b)$. Their generic nonzero denominators determine on which side the branches occur; existence alone does not make them stable.
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