= Solution
The nonzero <equilibrium points> are paired by reflection:
$$
x_*=\pm\sqrt{3(1-c/d)},\qquad y_*=0,\qquad z_*=(c/d)x_*,\qquad 0<c<d.
$$
At either <equilibrium point>, the derivative of the second equation with respect to $x$ is $-1+x_*^2=2-3c/d$. The <characteristic polynomial> becomes
$$
p_*(m)=m^3+(2-d/2)m^2+(3c/d-d-2)m+(d-c).
$$
Its constant term is nonzero away from the <pitchfork bifurcation>, so there is no further steady local bifurcation. At a <Hopf bifurcation> the condition $A_1A_2=A_3$ gives
$$
\boxed{c_H^*(d)={d(8+4d-d^2)\over12-d},\qquad0<d<1,\qquad
\omega^2={2d(1-d)\over12-d}.}
$$
Indeed $\tau=2-d/2$ and $\tau\omega^2=d-c>0$ require $d<4$; within this range $\omega^2>0$ is equivalent to $d<1$. The apparent singularity at $d=12$ does not hide another solution of the coefficient equation. The <Routh-Hurwitz stability criterion> shows that both <equilibrium points> are stable for $0<d<1$, $c_H^*(d)<c<d$, and unstable for $c<c_H^*(d)$. For $d\ge1$, $3c/d-d-2<1-d\le0$, so no nonzero <equilibrium point> is stable.
Their <Hopf bifurcations> are supercritical. Here is a coefficient check that includes the effect of the quadratic term around $x_*$. Let $B_2(u,v)=2x_*u_xv_x$, $C_2(u,v,w)=2u_xv_xw_x$ and normalize $q_x=1$ as before. The inverse-linearization term has $(L^{-1}B(q,\bar q))_x=3/x_*$. Also
$$
G_2={2i\omega-d/2\over-3\omega^2(\tau+2i\omega)},\qquad
((2i\omega I-L)^{-1}B(q,q))_x=2x_*G_2.
$$
The resonant cubic Hopf coefficient is
$$
l_1={1\over2\omega}\operatorname{Re}\{G[-10+4x_*^2G_2]\}
=-{8[d(4-d)^2+(28d+48)\omega^2]\over d\omega[(4-d)^2+4\omega^2][(4-d)^2+16\omega^2]}<0.
$$
In this simplification $x_*^2/\omega^2=3\tau/d$. Thus stable small <periodic orbits> appear around each nonzero <equilibrium point> on the $c<c_H^*(d)$ side. This classifies the local branches; the sketch does not infer uncomputed distant global bifurcations.
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