  We comment on some peculiarities of matter with and without Weyl invariance
coupled to classical $2d$ Einstein-Hilbert gravity for several models, in
particular, related to the counting of degrees of freedom and on the dynamics.
We find that theories where the matter action is Weyl invariant has generically
more degrees of freedom than action without the invariance. This follows from
the Weyl invariance of the metric equations of motion independently of the
invariance of the action. Then, we study another set of models with scalar
fields and show that solutions to the equations of motion are either trivial or
inconsistent. To our knowledge, these aspects of classical $2d$ gravity have
not been put forward and can be interesting to be remembered when using it as a
toy model for $4d$ gravity. The goal of this note is also as a pedagogical
exercise: our results follow from standard methods, but we emphasize more
direct computations.
