  Having in mind extensions of 2D holography beyond the Jackiw-Teitelboim model
we propose holographic counterterms and asymptotic conditions for a family of
asymptotically AdS$_2$ dilaton gravity models leading to a consistent
variational problem and a finite on-shell action. We show the presence of
asymptotic Virasoro symmetries in all these models. The Schwarzian action
generates (a part) of the equations of motion governing the asymptotic degrees
of freedom. We also analyse the applicability of various entropy formulae. By a
dilaton-dependent conformal transformation our results are extended to an even
larger class of models having exotic asymptotic behavior. We also analyse
asymptotic symmetries for some other classes of dilaton gravities without,
however, constructing holographic counterterms.
