Symmetry reduction simplifies a PDE by reducing the number of independent variables — use a translation symmetry to go from 2+1 to 1+1 dimensions, for example. When the symmetry preserves a geometric structure (a Poisson bracket, a variational principle), the reduced system inherits that structure. But what if the symmetry rescales the structure instead of preserving it?
The paper handles this case with a shift rule: when a symmetry X reduces the system and another symmetry X_s rescales the geometric structure, the restriction of X_s to the reduced system is shifted — it acts differently than the naive restriction would suggest. The shift is computable from the commutation relations between X and X_s.
Two phenomena emerge that don't occur in the structure-preserving case. Emergence of invariance: a geometric structure that is not invariant under a symmetry in the full system can become invariant under the restricted symmetry in the reduced system. The rescaling and the restriction can cancel. Loss of invariance: the reverse — a structure invariant in the full system can lose invariance upon reduction, because the shift introduces a non-trivial action where there was none.
Applied to the Lin-Reissner-Tsien equation of transonic gas flows, the framework produces exact closed-form solutions validated numerically. The Boussinesq system demonstrates inherited Poisson brackets — the algebraic structure survives reduction but is modified by the shift.
Symmetry that changes the playing field. Reduction doesn't just simplify the system — it creates and destroys structural properties. The shift rule tracks exactly what survives.