Invariant reduction for PDEs. II: The general mechanism

Given a local (point, contact, or higher) symmetry of a system of partial differential equations, one can consider the system that describes the invariant solutions (the invariant system). It seems natural to expect that the invariant system inherits symmetry-invariant geometric structures in a specific way. We propose a mechanism of reduction of symmetry-invariant geometric structures, which relates them to their counterparts on the respective invariant systems. This mechanism is homological and covers the stationary action principle and all terms of the first page of the Vinogradov C-spectral sequence. In particular, it applies to invariant conservation laws, presymplectic structures, and internal Lagrangians. A version of Noether's theorem naturally arises for systems that describe invariant solutions. Furthermore, we explore the relationship between the C-spectral sequences of a system of PDEs and systems that are satisfied by its symmetry-invariant solutions. Challenges associated with multi-reduction under non-commutative symmetry algebras are also clarified.