Origins and symmetry interpretation of non-Casimir integrals in IL^{(0,1)}QG

Establish the precise origin of the non-Casimir integrals of motion in the IL^{(0,1)}QG model—specifically, derive and prove their symmetry-based interpretation via an appropriate Noether theorem (e.g., particle relabeling symmetry in the Euler–Poincaré/Lie–Poisson formulations), thereby clarifying why these invariants arise despite not being Casimirs of the full Lie–Poisson bracket.

Background

Beyond the standard Casimirs of the IL^{(0,1)}QG Lie–Poisson bracket, the model possesses additional integrals of motion (the weak Casimirs) that neither commute with all functionals of the full bracket nor are obviously tied to explicit Eulerian symmetries.

The authors’ preliminary work suggests these invariants may stem from particle relabeling symmetry and could be obtained via a Noether-type argument in the parent primitive-equation framework, but a rigorous derivation and symmetry identification for IL^{(0,1)}QG is left open.