We construct a hierarchy of integrable systems whose Poisson structure
corresponds to the BMS3 algebra, and then discuss its description in terms
of the Riemannian geometry of locally flat spacetimes in three dimensions. The
analysis is performed in terms of two-dimensional gauge fields for isl(2,R).
Although the algebra is not semisimple, the formulation can be carried out \`a
la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear
metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative
integer k, and defined through a suitable generalization of the Gelfand-Dikii
polynomials. The symmetries of the hierarchy are explicitly found. For k≥1, the corresponding conserved charges span an infinite-dimensional Abelian
algebra without central extensions, and they are in involution; while in the
case of k=0, they generate the BMS3 algebra. In the special case of
k=1, by virtue of a suitable field redefinition and time scaling, the field
equations are shown to be equivalent to a specific type of the Hirota-Satsuma
coupled KdV systems. For k≥1, the hierarchy also includes the so-called
perturbed KdV equations as a particular case. A wide class of analytic
solutions is also explicitly constructed for a generic value of k.
Remarkably, the dynamics can be fully geometrized so as to describe the
evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed,
General Relativity in 3D can be endowed with a suitable set of boundary
conditions, so that the Einstein equations precisely reduce to the ones of the
hierarchy aforementioned. The symmetries of the integrable systems then arise
as diffeomorphisms that preserve the asymptotic form of the spacetime metric,
and therefore, they become Noetherian. The infinite set of conserved charges is
recovered from the corresponding surface integrals in the canonical approach.Comment: 34 pages, 2 figure