  This work introduces a novel family of boundary conditions for AdS$_3$
General Relativity, constructed through a polynomial expansion in negative
integer powers of the Brown-Henneaux central charge. The associated dynamics is
governed by the Dym hierarchy of integrable equations. It is shown that the
infinite set of Dym conserved charges generates an abelian asymptotic symmetry
group. Additionally, these boundary conditions encompass black hole solutions,
whose thermodynamic properties are examined.
