= Solution
For $\nu=1$, the <equilibria> are $(0,0)$ and $(\pm1,0)$. At the origin the <eigenvalues> are $1,-1$, so it is a saddle for every $\kappa$.
At either nonzero point the <Jacobian matrix> is $\left(\begin{smallmatrix}0&1\\-2&\kappa\end{smallmatrix}\right)$, with <eigenvalues>
$$
\boxed{\lambda_\pm=\frac{\kappa\pm\sqrt{\kappa^2-8}}2.}
$$
Both points are attracting for $\kappa<0$ and repelling for $\kappa>0$. For $-2\sqrt2<\kappa<0$ they are <stable focus>[stable foci]; for $\kappa<-2\sqrt2$ <stable nodes>. For $0<\kappa<2\sqrt2$ they are <unstable focus>[unstable foci]; for $\kappa>2\sqrt2$ <unstable nodes>. At the two repeated-root boundaries they are <degenerate nodes>. At $\kappa=0$, <linearization> has purely imaginary <eigenvalues>; the conserved <energy> in the next part shows that they are actual nonlinear <centre equilibrium>[centres]. Thus the zero-real-part case is resolved by an additional <first integral>, rather than by the linear test alone.
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