Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 1

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

Suppose that $ \displaystyle \frac{ \partial \Phi }{ \partial x }=0$.

Then $\Phi$ is the first integral if its Poisson Bracket with the Hamiltonian vanishes.

Proof
The statement is proven by setting $ \displaystyle \frac{ \partial \Phi }{ \partial x }=0$ in the main result of Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation.