Strong symmetry

In the definition of a symmetry of a DE $\mathcal{S}$ we required that if $\mathrm{\Delta }$ is the function such that $\mathcal{S}=\{\mathrm{\Delta }=0\}$ and $X^{(n)}$ is the prolongation of (the generator of) the symmetry then

for every $p$ such that $\mathrm{\Delta }(p)=0.$

If equation (1) is satisfied for every $p$, the symmetry is called a strong symmetry.

There is a relation between usual symmetries and strong ones, the CDW theorem.