  In this work, we consider a two-dimensional (2D) dilaton gravity model where
the dilaton kinetic term $\mathcal{X}$ is modified by an additional derivative
coupling term $\alpha\mathcal{X}^2$. In the case with a canonical scalar matter
field, the field equations of this model have a simple first-order formalism,
from which exact static kink solutions can be constructed. The novelty of these
solutions is that the corresponding metric can be asymptotically flat rather
than asymptotically anti de Sitter. The linear stability and the localization
of scalar matter fields are also studied. It was found that the solutions are
stable against small linear perturbations, and the localization of scalar
matter fields can be realized by introducing scalar-kink interactions.
