= Solution
Within the positive-amplitude wedge the reduced <periodic orbit> equilibrium has amplitude <trace> $\mu_1$ and positive <determinant> $4r_*^2$. At $\mu_1=0,\mu_2<0$ the transverse pair is imaginary, so the possible secondary event is a <Hopf bifurcation> of the amplitude equations, corresponding to an invariant <torus> in the full system.
It is degenerate. Set $\mu_2=-a^2$ and $\mu_1=0$. The <first integral of the quadratic fold-Hopf amplitude flow> is
$$
F=r(a^2-r^2/3-x^2),\qquad F_r=a^2-r^2-x^2,\qquad F_x=-2rx.
$$
The amplitude equations have the <Hamiltonian system> form $\dot r=F_x$, $\dot x=-F_r$, so $\dot F=0$. At $(r,x)=(a,0)$, $F$ has a strict local maximum $2a^3/3$ and negative-definite Hessian. Every nearby regular level is a closed amplitude <periodic orbit>, rather than one isolated orbit selected by nonlinear damping. Adding the rotating phase gives an entire family of invariant <tori>; the full motion is quasiperiodic or periodic according to the phase advance during one amplitude period.
For $\mu_1\ne0$, the amplitude <vector field> has constant <divergence> $\mu_1$. The <Bendixson-Dulac theorem> rules out closed amplitude trajectories in the physical simply connected half-plane $r>0$. Hence there is no ordinary branch of isolated invariant <tori> on either side in this quadratic system. The secondary event has vanishing radial saturation coefficients; its imaginary linear pair alone does not establish a nondegenerate torus bifurcation.
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