= Solution
The <matrix> $N$ is a <skew-symmetric matrix>, so $x^TNx=0$ for real $x$. Therefore the <energy-neutral rotational nonlinearity> contributes nothing directly to the energy balance:
$$
\boxed{\frac{dE}{dt}=x^T\dot x=|x|x^TNx+x^TLx=x^TLx
=-\frac{x_1^2}{R}+x_1x_2-\frac{x_2^2}{4R},\qquad R=\mathrm{Re}.}
$$
The nonlinear term rotates the state without changing its <Euclidean norm>. It can still change the orientation, and thereby affect later energy growth through the linear term. Thus energy cancellation does not mean the nonlinear and linear trajectories have identical energy histories.
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