  We study static kink solutions in a generalized two-dimensional dilaton
gravity model, where the kinetic term of the dilaton is generalized to be an
arbitrary function of the canonical one $\mathcal X= -\frac12 (\nabla
\varphi)^2$, say $\mathcal F(\mathcal X)$, and the kink is generated by a
canonical scalar matter field $\phi$. It is found that for arbitrary $\mathcal
F(\mathcal X)$, the background field equations have a simple first-order
formalism, and the linear perturbation equation can always be written as a
Schr\"odinger-like equation with factorizable Hamiltonian operator. After
choosing appropriate $\mathcal F(\mathcal X)$ and superpotential, we obtain a
sine-Gordon type kink solution with pure AdS$_2$ metric. The linear
perturbation issue of this solution becomes an exactly solvable conformal
quantum mechanics problem, if one of the model parameter takes a critical
value.
