Nonlinear centres with conserved quantities

Consider the differential system $˙ 𝑥 = 𝑓 (𝑥)$ with differentiable $𝑓 : ℝ 𝑛 \to ℝ 𝑛$ . Suppose there exists a conserved quantity $𝐸 (𝑥)$ and that $𝑥 *$ is an isolated fixed point. If $𝑥 *$ is a local extremum of $𝐸$ , then there exists a neighbourhood of $𝑥 *$ containing only orbits.¹ #m/thm/dynamics/flow