I'm looking for "famous" or otherwise well-known 2d Riemannian manifolds which have non-constant curvatures but have a non-trivial Killing vector field. Of course there are tons of spaces like these, for instance if we parametrize the plane (or a subset of it) by $(r,\phi)$ then any conformal rescaling of the flat metric by a conformal factor which only depends on $r$ will be generally good, i.e. have non-constant curvature but the rotations generated by $\partial/\partial\phi$ will still be a symmetry.

Are there special spaces which are somehow famous or well-studied because of some special property? Ideally, I'm looking for deformations of the Poincare disc.