Symmetries of PDE systems and correspondences between jet spaces

Let M be a manifold. A PDE system can be prolonged to another one R⁎⊆T⁎M (Jiménez et al. (2005) [10]). In analogy with the higher-order symmetries, symmetries of R⁎ will be called higher-dimensional symmetries of R. For a broad class of PDE systems we prove that every (infinitesimal or finite) symmetry of R comes from another one of R⁎. We show that R⁎ does not have internal (infinitesimal) symmetries (modulo trivial symmetries). This fact allows us, in the infinitesimal case, to compute the internal symmetries of R as external symmetries of R⁎. We also give an algorithmic method to obtain solutions of R invariant by a given internal symmetry.