Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 2

Theorem
Consider the Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation.

Let $\Phi = H$.

Let $\dfrac {\partial H} {\partial x} = 0$.

Then $H$ is the first integral of Euler's Equations.

Proof
The statment is proven from Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation

by setting $\Phi = H$ and $\dfrac {\partial H} {\partial x} = 0$, and noticing that $\left[{H, H}\right] = 0$.