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:

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

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

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