$$\left \{ \begin{array}{l} x'(t) = y(t)\\ y'(t) = -W'(x(t)) \end{array} \right.$$ where $W : \mathbb R^2 \to \mathbb R^+, (x, y) \mapsto \frac{y^2}{2} + W(x)$ a $\mathcal C^2$ function and it's the first integral of the system.
The equilibrium points are $E = \{ (x, y)\in \mathbb R^2 \ | \ y = 0 \text{ and } W'(x) = 0\}$
Suppose $\forall (x, y) \in E, W''(x) \ne 0$, show that $E$ is discret.
Suppose $E$ has a limit point $x_0$ which is finite, namely there exists $\{x_n\}\subset E$ such that $\lim_{n\to\infty}x_n=x_0$. Then $$ W'(x_n)=0, \forall n\in N $$ from which one has $$ \frac{W'(x_m)-W'(x_n)}{x_m-x_n}=0, \forall m,n\in N.$$ Letting $m,n\to\infty$ gives $$ W''(x_0)=0$$ which is against the assumption $W''(x)\neq 0$ for all $x\in E$.