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

Corollary to Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation
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 $\left[{H, H}\right] = 0$.