= Solution
The <equilibria> have $y=0$ and $x(\lambda+x^2)=0$. Thus
$$
\boxed{O=(0,0)\text{ for all parameters},\qquad P_\pm=(\pm\sqrt{-\lambda},0)\text{ for }\lambda<0.}
$$
At $O$, the <linearization> has <trace> $\kappa$ and <determinant> $\lambda$. Hence $O$ is a <saddle equilibrium> when $\lambda<0$, a <sink equilibrium> when $\lambda>0,\kappa<0$, and a <source equilibrium> when $\lambda>0,\kappa>0$. The <sink equilibrium>/<source equilibrium> may be a <node equilibrium> or a <focus equilibrium>; the node-focus discriminant is not a <bifurcation> curve. At $P_\pm$, the <trace> and <determinant> are $\kappa+\lambda$ and $-2\lambda$. These points are <sink equilibria> for $\kappa<-\lambda$ and <source equilibria> for $\kappa>-\lambda$.
The local <bifurcation> curves are therefore
$$
\boxed{\lambda=0\quad\text{(pitchfork)},\qquad
\kappa=0,\ \lambda>0\quad\text{(Hopf at }O),\qquad
\kappa=-\lambda,\ \lambda<0\quad\text{(Hopf at }P_\pm).}
$$
For $\kappa\ne0$ on the pitchfork curve, a leading centre-manifold equation is $\dot x=(\lambda x+x^3)/\kappa+\cdots$. For $\kappa<0$, with unfolding parameter $-\lambda$, the pitchfork is supercritical and creates two <sink equilibria> from a <sink equilibrium> that becomes a <saddle equilibrium>. For $\kappa>0$ it is subcritical in the centre direction; the nonzero branches are unstable, and the transverse <eigenvalue> is also positive. The origin of parameter space is a reflection-symmetric double-zero degeneration.
To determine Hopf criticality at the origin, use the <energy> $E=y^2/2+\lambda x^2/2+x^4/4$, whose <derivative> is $(\kappa-x^2)y^2$. For a small nearly harmonic oscillation $x=R\cos(\sqrt\lambda\,t)$, averaging gives $\dot R=\kappa R/2-R^3/8+\cdots$. Thus this <Hopf bifurcation> is supercritical and creates an attracting <periodic orbit> for $\kappa>0$. By the supplied opposite-criticality fact, the simultaneous <Hopf bifurcations> at the two nonzero <equilibria> are subcritical. A direct calculation confirms this: writing $x=\sqrt{-\lambda}+X$, $Y=-y/\sqrt{-2\lambda}$ puts the cubic radial coefficient at $1/4>0$, including the contribution from the quadratic terms.
The figure includes these local curves and the global curves derived below. All lines emerging from the double-zero point are local asymptotic sketches.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-56-symmetric-parameters.png]
{title=Local and global curves near the symmetric double-zero point}
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