= Solution
Write $Q=x^2y^2+y^2z^2+z^2x^2$ and $V=xyz$. Direct differentiation of the cubic <amplitude equations> gives
$$
\dot X=2\mu X-2aX^2+2(e-c)Q,\qquad
\dot V=V\bigl(3\mu-(3a+c-e)X\bigr).
$$
When $c=e$, these reduce to $\dot X=2X(\mu-aX)$ and $\dot V=3V(\mu-aX)$. Therefore, wherever $X>0$,
$$
\boxed{\frac d{dt}\left(\frac{V}{X^{3/2}}\right)=0.}
$$
This conserved directional quantity is nonconstant in every neighborhood of a nonzero <equilibrium>. If such an <equilibrium> were <asymptotically stable>, all sufficiently nearby initial points would approach it, forcing their conserved values to equal its value. That is impossible. The origin is unstable since its linearization is $\mu I$ with $\mu>0$. Hence \b[there are no <asymptotically stable> <equilibria> when $c=e$] in the displayed cubic system. For $c=e=0$ the sphere is entirely <equilibria>, so its individual points still cannot be <asymptotically stable>.
The radial equation attracts every nonzero radius toward $X=R^2=\mu/a$. On this invariant sphere, with $c=e$, the remaining motion is
$$
\dot x=cx(z^2-y^2),\quad
\dot y=cy(x^2-z^2),\quad
\dot z=cz(y^2-x^2),\qquad V=\text{constant}.
$$
For $c\ne0$, the six coordinate-axis points are saddles on the sphere. The eight equal-magnitude points $|x|=|y|=|z|=R/\sqrt3$ are centers of the restricted motion. In each open octant the regular level curves of $V$ are closed loops around that octant's center. The coordinate planes, where $V=0$, form <heteroclinic orbits> joining the axis saddles. In the positive octant, for $c>0$ the boundary orientation is $x\text{-axis}\to y\text{-axis}\to z\text{-axis}\to x\text{-axis}$; for $c<0$ all arrows reverse. Sign reflections give the other octants.
The following sketch uses normalized squared coordinates $U=x^2/R^2$, $W=y^2/R^2$, $Z=z^2/R^2$, with $U+W+Z=1$. This is a one-to-one coordinate representation of each closed spherical octant, so its triangle depicts the sphere's trajectories without hiding a hemisphere. The center is $U=W=Z=1/3$, and the interior loops are contours of $UWZ$. This is the <conserved octant dynamics of a cyclic amplitude system>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-78-sphere.png]
{title=Phase portrait of one octant of the invariant sphere in normalized squared coordinates, with closed orbits and a heteroclinic boundary cycle}
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