 Prolongation of a vector field

For $V=\xi (x,u){\partial }_x+\eta (x,u){\partial }_u$, its prolongation to derivatives through order $n$ isIt generates a symmetry of $\Delta =0$ exactly when ${\mathrm{pr}}^{(n)}V(\Delta )$ vanishes on the equation manifold.

Prolongation lifts a vector field to the jet bundle of maps so that its flow acts on derivatives of the dependent variables as well as on their values.