  We present a new set of boundary conditions for General Relativity on
AdS$_3$, where the dynamics of the boundary degrees of freedom are described by
two independent left and right members of the Gardner hierarchy of integrable
equations, also known as the "mixed KdV-mKdV" hierarchy. This integrable system
has the very special property that simultaneously combines both, the
Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies in a
single integrable structure. This relationship between gravitation in
three-dimensional spacetimes and two-dimensional integrable systems is based on
an extension of the recently introduced "soft hairy boundary conditions" on
AdS$_3$, where the chemical potentials are now allowed to depend locally on the
dynamical fields and their spatial derivatives. The complete integrable
structure of the Gardner system, i.e., the phase space, the Poisson brackets
and the infinite number of commuting conserved charges, are directly obtained
from the asymptotic analysis and the conserved surface integrals in the
gravitational theory. These boundary conditions have the particular property
that they can also be interpreted as being defined in the near horizon region
of spacetimes with event horizons. Black hole solutions are then naturally
accommodated within our boundary conditions, and are described by static
configurations associated to the corresponding member of the Gardner hierarchy.
The thermodynamic properties of the black holes in the ensembles defined by our
boundary conditions are also discussed. Finally, we show that our results can
be naturally extended to the case of a vanishing cosmological constant, and the
integrable system turns out to be precisely the same as in the case of AdS$_3$.
