Definition:First Integral of System of Differential Equations

Definition
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= f\left({ \langle x_i \rangle_{1 \le i \le n}, \langle g^{ \left({ j } \right) } \left({ \langle x_i \rangle_{1 \le i \le n} } \right) \rangle_{ 0 \le j \le k} } \right), \quad {n, k} \in \N$

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

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