Main idea: The paper revisits a previously proposed integrable system connected to BMS3, the symmetry group that describes allowed changes at the edge of three‑dimensional, asymptotically flat spacetimes. The authors build new, systematic mathematical constructions that show the hierarchical family of partial differential equations is integrable by several structural methods. They also compare the flat‑space case to the anti‑de Sitter (AdS3) case and explore spectral problems tied to certain Schrödinger operators.

What the researchers did: Starting from algebraic structures behind BMS3, the authors construct a bi‑Hamiltonian hierarchy. “Bi‑Hamiltonian” means the same dynamics can be written with two compatible Hamiltonian (energy‑like) structures, which is a common signature of integrability. They obtain this hierarchy in two ways. One route uses the variational complex on a ring of polynomial symbols and produces a Nijenhuis operator (a tool that helps generate the sequence of commuting flows). The other route uses the Lie‑Poisson bracket of the BMS3 algebra together with a suitable “frozen” partner bracket. They repeat the strategy for the AdS3 algebra and check that taking the flat limit connects the AdS3 construction back to the BMS3 case.

How it works, at a high level: The bi‑Hamiltonian setup gives a recursive mechanism to build many commuting evolution equations and conserved quantities. Attaching a Nijenhuis operator provides a systematic map between these conserved quantities and the flows they generate. The paper also gives a different description in terms of a τ (tau)‑scheme, and it explains how coadjoint orbits of BMS3 — geometric spaces that encode the algebra’s action on its dual — serve as the Lax phase space for a subclass of energy‑dependent Schrödinger‑type spectral problems. A Lax description is another standard way to show integrability, by encoding dynamics as compatibility conditions of linear problems.