Symmetries of PDE systems and correspondences between jet spaces

Let M be a manifold. A PDE system R ⊆ J m 1 M 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.