  We review classical approaches to the problem of isometrically embedding a
Riemannian surface into Euclidean 3-space, including coordinate-based
approaches exposited by Darboux and Eisenhart, as well as the moving-frames
based approaches advocated by Cartan. In particular, the first approach
involves reducing the problem to solving a single PDE; settling the question of
when this PDE is integrable by the method of Darboux is the subject of this
short note. This is surprisingly easy, since it is related to the analogous
question for the isometric embedding system arising from the moving frames
approach, and this was accomplished in recent joint work with Clelland, Tehseen
and Vassiliou (arXiv:1801.00241).
