Integrability in Asymptotic Symmetries of Spacetime: the ${\text{BMS}}_3$ scenario

We discuss the existence of a well-defined ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$-integrable hierarchy, introduced in Fuentealba et al. (JHEP, 2018), using different structural methodologies. Specifically, from the variational complex on the ring of polynomial symbols, a ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$ bi-Hamiltonian structure is constructed on which a Nijenhuis operator can be attached. A similar argument is made for the ${\mathrm{A}\mathrm{d}\mathrm{S}}_3$ case, where we check that the flat limit of the latter recovers the asymptotically flat situation. An alternative $\tau$-scheme description is also presented. Moreover, a Lie-Poisson description suggests that such a ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$-hierarchy is not unique. For a subclass of so-called energy-dependent Schr"odinger operators, it is shown that their Lax flows are described by the coadjoint orbits of ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$.