Partial Derivatives of Solution of Hamilton-Jacobi Equation are First Integrals of Euler's Equations

Theorem
Let $ \mathbf y= \langle y_i \rangle_{1 \le i \le n}$, $ \boldsymbol \alpha= \langle \alpha_i \rangle_{1 \le i \le m}$ be vectors, where $n \le m$.

Let $ S= S \left({ x, \mathbf y, \boldsymbol \alpha } \right)$ be a solution of Hamilton-Jacobi Equation, where $ \boldsymbol \alpha$ are parameters.

Then each derivative


 * $ \displaystyle \frac{ \partial S}{ \partial \alpha_i}$

is a first integral of canonical Euler's equations.