= Solution
The Lorenz equilibria satisfy $y=x$, $z=x^2/b$ and $x(r-1-z)=0$. Thus
$$
\boxed{O=(0,0,0),\qquad C_\pm=(\pm\sqrt{b(r-1)},\pm\sqrt{b(r-1)},r-1)\quad(r>1).}
$$
At the origin the third <eigenvalue> is $-b$ and the other two solve $\lambda^2+(\sigma+1)\lambda+\sigma(1-r)=0$. Hence the origin is stable for $r<1$, has one zero <eigenvalue> at $r=1$, and becomes a <saddle equilibrium> with one unstable direction for $r>1$. The <symmetry> $(x,y,z)\mapsto(-x,-y,z)$ interchanges the new branches.
The nonzero-equilibrium characteristic polynomial is
$$
\boxed{P(\lambda)=\lambda^3+(\sigma+b+1)\lambda^2+b(\sigma+r)\lambda+2\sigma b(r-1).}
$$
Near $r=1$ its Routh-Hurwitz inequalities are strict, so both new branches are stable. More explicitly, center-manifold projection at $r=1$ gives
$$
\dot q=\frac\sigma{\sigma+1}(r-1)q-\frac\sigma{b(\sigma+1)}q^3+
O((r-1)^2q,(r-1)q^3,q^5).
$$
The linear coefficient crosses positively and the cubic coefficient is negative. Therefore \b[the equilibrium <bifurcation> is a supercritical pitchfork at $r=1$], not a <saddle equilibrium>-node.
For completeness, the later stability boundary satisfies
$$
(\sigma+b+1)(\sigma+r)=2\sigma(r-1).
$$
If $\sigma\leq b+1$, the nonzero branches remain linearly stable for all $r>1$. If $\sigma>b+1$, they lose stability at
$$
\boxed{r_H=\frac{\sigma(\sigma+b+3)}{\sigma-b-1},\qquad
\omega_H^2=b(\sigma+r_H).}
$$
At that point $P=(\lambda+\sigma+b+1)(\lambda^2+\omega_H^2)$ and
$$
\left.\frac{d\operatorname{Re}\lambda}{dr}\right|_H
=\frac{b(\sigma-b-1)}{2[\omega_H^2+(\sigma+b+1)^2]}>0.
$$
This establishes a genuine transverse Hopf crossing. It is \b[subcritical] for these positive <parameters>, so it is not the supercritical Hopf <normal form of a dynamical system> from (c). One way to check its nonlinear sign is the cubic Hopf formula. At $C_+$ write $s=\sqrt{b(r_H-1)}$ and
$$
A=\begin{pmatrix}-\sigma&\sigma&0\\1&-1&-s\\s&s&-b\end{pmatrix},\qquad
B(v,w)=(0,-v_1w_3-v_3w_1,v_1w_2+v_2w_1).
$$
For right and adjoint eigenvectors $Aq=i\omega_Hq$, $A^*p=-i\omega_Hp$, $\langle p,q\rangle=1$, the cubic coefficient has the sign of
$$
\operatorname{Re}\langle p,
B(\bar q,(2i\omega_HI-A)^{-1}B(q,q))
-2B(q,A^{-1}B(q,\bar q))\rangle.
$$
Substitution and clearing positive factors reduces its sign to that of the following polynomial in $h=\sigma-b-1>0$:
$$
9h^4+2(9b+28)h^3+4(5b^2+29b+31)h^2+
4(b+1)(b+2)(5b+14)h+8(b+1)^2(b+2)^2.
$$
Every term is positive. The radial cubic is therefore positive, giving the unstable small cycle on the stable-equilibrium side. There is no <saddle equilibrium>-node among the equilibrium branches for $\sigma,b,r>0$: the only zero-<eigenvalue> branch intersection is the <symmetry>-forced pitchfork. This does not rule out separate global <bifurcations> of <periodic orbits>.
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