See [Gaeta 2020] remark 5.

Roughly speaking, el CDW theorem says that, up to cohomological considerations, if a system of DEs $\mathrm{\Delta }$ admits $X$ as a Lie symmetry there is always a system of DEs $\tilde{\mathrm{\Delta }}$ which admits $X$ as a strong symmetry and such that $\mathrm{\Delta }$ and $\tilde{\mathrm{\Delta }}$ have the same set of solutions.