  We consider different sets of AdS$_2$ boundary conditions for the
Jackiw-Teitelboim model in the linear dilaton sector where the dilaton is
allowed to fluctuate to leading order at the boundary of the Poincar\'e disk.
The most general set of boundary condtions is easily motivated in the gauge
theoretic formulation as a Poisson sigma model and has an $\mathfrak{sl}(2)$
current algebra as asymptotic symmetries. Consistency of the variational
principle requires a novel boundary counterterm in the holographically
renormalized action, namely a kinetic term for the dilaton. The on-shell action
can be naturally reformulated as a Schwarzian boundary action. While there can
be at most three canonical boundary charges on an equal-time slice, we consider
all Fourier modes of these charges with respect to the Euclidean boundary time
and study their associated algebras. Besides the (centerless)
$\mathfrak{sl}(2)$ current algebra we find for stricter boundary conditions a
Virasoro algebra, a warped conformal algebra and a $\mathfrak{u}(1)$ current
algebra. In each of these cases we get one half of a corresponding symmetry
algebra in three-dimensional Einstein gravity with negative cosmological
constant and analogous boundary conditions. However, on-shell some of these
algebras reduce to finite-dimensional ones, reminiscent of the on-shell
breaking of conformal invariance in SYK. We conclude with a discussion of
thermodynamical aspects, in particular the entropy and some Cardyology.
