  The use of a diffeomorphism covariant star product enables us to construct
diffeomorphism invariant gravities on noncommutative symplectic manifolds
without twisting the symmetries. As an example, we construct noncommutative
deformations of all two-dimensional dilaton gravity models thus overcoming some
difficulties of earlier approaches. One of such models appears to be
integrable. We find all classical solutions of this model and discuss their
properties.
