See [Gaeta 2020] remark 5.

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

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es