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

Theorem
Suppose $ \Phi= H$ and $ \displaystyle \frac{ \partial H }{ \partial x }=0$.

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

Proof
The statment is proven from by setting $ \Phi= H$ and $ \displaystyle \frac{ \partial H }{ \partial x }=0$, and noticing that $\left[{ H, H } \right]=0$.


 * : $\S 4.17$: First Integrals of the Euler Equations