= Solution
Put $S=u^2+v^2$. At a nonzero <equilibrium point> the two equations form the homogeneous linear system
$$
\begin{pmatrix}\mu-S/2&1-\sigma+S/2\\1+\sigma-S/2&\mu-S/2\end{pmatrix}\binom uv=0.
$$
Its <determinant> must vanish, giving $S^2/2-(\mu+\sigma)S+\mu^2+\sigma^2-1=0$. Thus
$$
\boxed{S=\mu+\sigma\pm\sqrt{2-(\mu-\sigma)^2},\quad S>0.}
$$
Each positive real <polynomial root> gives an antipodal <equilibrium point> pair; a negative <polynomial root> does not describe a real <equilibrium point>. A direction is specified by $\sin2\theta=S/2-\mu$, $\cos2\theta=S/2-\sigma$, where $u=\sqrt S\cos\theta$, $v=\sqrt S\sin\theta$. This is the <polar form of the cubic confinement model>. The origin is always an <equilibrium point>.
The origin's <eigenvalues> are $\lambda_\pm=\mu\pm\sqrt{1-\sigma^2}$. A simple zero <eigenvalue> occurs on $\mu^2+\sigma^2=1$, except at $\mu=0$. The symmetry of an <odd function> $(u,v)\mapsto(-u,-v)$ makes the generic steady bifurcation a <pitchfork bifurcation>. At a point $(\sigma,\mu_0)$ on this circle with $\mu_0\ne0$, vary $\mu=\mu_0+\delta$. Expanding the <radius> equation gives $S\sim2\mu_0\delta/(\mu_0+\sigma)$. In a signed, unit-eigenvector <centre manifold> coordinate, the cubic coefficient therefore has sign $-(\mu_0+\sigma)/(2\mu_0)$.
On the lower semicircle $\mu_0=-\sqrt{1-\sigma^2}$ the transverse <eigenvalue> is negative: the <pitchfork bifurcation> is supercritical for $\sigma<1/\sqrt2$ and subcritical for $\sigma>1/\sqrt2$. On the upper semicircle it is supercritical for $\sigma>-1/\sqrt2$ and subcritical for $\sigma<-1/\sqrt2$ in the <centre manifold> direction, but its transverse <eigenvalue> is positive, so the bifurcating <equilibrium points> are not fully attracting. At $(\sigma,\mu_0)=(\pm1/\sqrt2,\mp1/\sqrt2)$ the cubic vanishes. The <radius> equation instead gives $\delta=-S^2/(4\mu_0)+o(S^2)$, a nonzero quintic degeneracy with <amplitude> proportional to $|\delta|^{1/4}$: these are <degenerate pitchforks in the cubic confinement model>.
A <Hopf bifurcation> occurs at $\mu=0$, $|\sigma|>1$, with frequency $\sqrt{\sigma^2-1}$. To determine its direction, first note that the divergence is $2\mu-2S$. The <Bendixson-Dulac criterion> excludes every nonconstant <periodic orbit> for $\mu\le0$, since this divergence is negative except possibly at the origin. We can also calculate a nonzero saturation coefficient. Set $a=|\sigma+1|$, $b=|\sigma-1|$ and $V=au^2+bv^2$. At $\mu=0$ the linear orbits are <ellipses> of constant $V$. Direct <differentiation> gives
$$
\dot V=2\mu V-S\bigl[V+(b-a)uv\bigr].
$$
On a linear <ellipse>, write $u=\sqrt{V/a}\cos\phi$, $v=\sqrt{V/b}\sin\phi$. The angular speed is constant; the averages of $u^3v$ and $uv^3$ vanish, while $\langle S\rangle=V|\sigma|/(\sigma^2-1)$. Averaging the cubic terms, or removing their oscillatory parts by a periodic change of coordinates in its <normal form>, gives for $\varrho=\sqrt V$
$$
\dot\varrho=\mu\varrho-\frac{|\sigma|}{2(\sigma^2-1)}\varrho^3
+O(\varrho^5+|\mu|\varrho^3).
$$
The cubic coefficient is strictly negative. Hence a small attracting <limit cycle> exists for $\mu>0$, and
$$
\boxed{\text{the Hopf bifurcation is supercritical}.}
$$
Finally $(\sigma,\mu)=(\pm1,0)$ have a nonzero <nilpotent matrix> as their <linearization> with two zero <eigenvalues>. They are reflection-symmetric double-zero bifurcations where the <Hopf bifurcation> and steady thresholds meet; an ordinary nonzero-frequency <Hopf bifurcation> calculation does not apply there. These, the two degenerate <pitchfork bifurcation> points, and the generic <pitchfork bifurcation>/<Hopf bifurcation> loci above exhaust the origin's local loss-of-hyperbolicity possibilities.
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