  We study the gauge theory formulation of Jackiw-Teitelboim gravity and
propose Korteweg-de Vries asymptotic conditions that generalize the asymptotic
dynamics of the theory. They permit to construct an enlarged set of boundary
actions formed by higher order generalizations of the Schwarzian derivative
that contain the Schwarzian as lower term in a tower of SL$(2,\mathbb{R})$
invariants. They are extracted from the KdV Hamiltonians and can be obtained
recursively. As a result, the conformal symmetry breaking observed in nearly
AdS$_{2}$ is characterized by a much larger set of dynamical modes associated
to the stationary KdV hierarchy. We study quantum perturbation theory for the
generalized Schwarzian action including the symplectic measure and compute the
one-loop correction to the partition function. We find that that despite the
non-linear nature of the higher-Schwarzian contribution, it acquires a
manageable expression that renders a curious leading temperature dependence on
the entropy $S=\#T^{a}$ for $a$ an odd integer.
