= Solution
A <centre equilibrium> of the linearized real planar system has
$$
\boxed{\lambda_1=i\omega,\qquad\lambda_2=-i\omega,\qquad\omega\in\mathbb R\setminus\{0\}.}
$$
After a real invertible <linear change of coordinates>, the system is a constant angular rotation. Its nonzero orbits are ellipses of period $2\pi/|\omega|$, so it is stable but not asymptotically stable. This is a classification of the \b[<linearization>]: <a centre in the linearization need not be a nonlinear centre>. In particular, imaginary <eigenvalues> alone do not prove a nonlinear <centre equilibrium>, which requires a punctured neighbourhood filled with <periodic orbits>. Part (c) illustrates exactly this limitation.
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