\begin{definition}[Definition:First Integral of System of Differential Equations]
Let $S$ be a system of differential equations.
Let $g$ be a function which satisfies $S$.
Let $f$ be a function.
{{explain|Establish the domain and codomain of these functions}}
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 $S$.
\end{definition}