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

For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic structures, variational principles, and Poisson brackets, are inherited by the systems governing symmetry-invariant solutions. We extend this mechanism to geometric structures that are not invariant but are $\textit{rescaled}$ by a symmetry. Specifically, if $X$ is the symmetry used for reduction, $X_s$ is a symmetry satisfying $[X_s,X]=aX$, and the Lie derivative $\mathcal{L}_{X_s}$ acts on an $X$-invariant element of the Vinogradov $\mathcal{C}$-spectral sequence as multiplication by $b$, then the restricted symmetry $X_s|_{\mathcal{E}_X}$ acts on the corresponding reduction as multiplication by $a+b$. This shift rule gives rise to two phenomena: the $\textit{emergence of invariance}$, where reductions acquire an invariance that was not present at the level of the original structure, and the $\textit{loss of invariance}$, where reductions of invariant structures are no longer invariant. As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs. The framework is illustrated by two examples: the Lin--Reissner--Tsien equation of potential nonstationary transonic gas flows, for which closed-form exact solutions are obtained and validated numerically, and the potential Boussinesq system, for which the inherited Poisson bracket is employed to describe solutions determined by algebraic equations.