Definition:First Integral of System of Differential Equations

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Let $S$ be a system of differential equations.

Let $g$ be a function, which satisfies $S$.

Let $f$ be a function.

Let $f$ depend on variables (denoted by ordered tuples here) of $S$ independently as well as through $g$ and its derivatives:

$ f=\map f{\langle x_i\rangle_{1\le i\le n},\langle g^{\paren j}\paren{\langle x_i\rangle_{1\le i\le n} }\rangle_{0\le j\le k} } , \quad {n,k}\in\N$

Suppose, there exists $g$ such that $f$ is a constant.

Then $f$ is the first integral of $S$.