Let $\mathbf F$ be a system of 1st order differential equations in $n>3$ variables $$\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^n$$ $$\frac{d\mathbf{u}}{dt} = \mathbf{F}(\mathbf{u})$$
such that $\frac{\partial\mathbf{F}}{\partial t} = 0$.
We've referenced $\mathbf{u} : \mathbb{R} \to \mathbb{R}^n$. Let's also name a function $\mathbf{v} : \mathbb{R} \to \mathbb{R}^3$.
$\mathbf{u}$ and $\mathbf{v}$ have multiple elements, each expressible as an independent function of $t$. Call the 0th elements $\mathrm{u}_0$ and $\mathrm{v}_0$.
Is it always possible to find a system $\mathbf G$ in three variables $$\frac{d\mathbf{v}}{dt} = \mathbf{G}(\mathbf{v})$$ such that $\frac{\partial\mathbf{G}}{\partial t} = 0$ and $\mathrm{u}_0 = \mathrm{v}_0$?
To keep things simple you may assume $\mathbf{F}$ is smooth. Is the answer very different if it's merely continuous?
Is there a more succinct way to ask this question?