Elementary first integrals of integral differential systems

For a field $F$ of characteristic zero with a derivation $\delta$, we provide a necessary and sufficient condition for a system of differential equations \begin{equation*} δy_1=f_0, \quadδy_2=f_1,\quad\dots\quad,δy_n=f_{n-1}, \end{equation*} where $f_0\in F$, $f_1\in F[y_1]$, $\dots$, $f_{n-1}\in F[y_1,\dots ,y_{n-1}]$, to have elementary first integrals.