Invariant Reduction for Partial Differential Equations. IV: Symmetries that Rescale Geometric Structures

This framework extends invariant reduction to geometric structures that are not invariant but are rescaled by a symmetry in partial differential equations. It describes how invariant geometric structures (conservation laws, presymplectic structures, variational principles, Poisson brackets) are inherited by symmetry-invariant solutions. As an application, it describes a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws, illustrated with the Lin-Reissner-Tsien equation (transonic gas flows) and the potential Boussinesq system, obtaining closed-form exact solutions validated numerically.