= Solution
For the linearized system a <centre equilibrium>[centre] occurs when $\boxed{\lambda_{1,2}=\pm i\beta,\ \beta\ne0}$. Its real solutions are periodic ellipses after an invertible linear coordinate change.
As explained by <a centre in the linearization need not be a nonlinear centre>, for a nonlinear system these <eigenvalues> alone do \b[not] guarantee a <centre equilibrium>[centre]. For example $\dot X=-Y-X(X^2+Y^2)$, $\dot Y=X-Y(X^2+Y^2)$ has <eigenvalues> $\pm i$ at zero but polar equations $\dot r=-r^3$, $\dot\theta=1$, so trajectories spiral inward. Removing the cubic terms gives a genuine <centre equilibrium>[centre] with the same Jacobian. Thus there is no <eigenvalue>-only sufficient condition for a nonlinear <centre equilibrium>[centre] in the stated generality.
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