  Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $(M,
\boldsymbol{g})$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian
manifold $(N, \boldsymbol{h})$, one can ask under what circumstances does the
exterior differential system $\mathcal{I}$ for the isometric embedding
$M\hookrightarrow N$ have particularly nice solvability properties. In this
paper we give a classification of all $2$-metrics $\boldsymbol{g}$ whose local
isometric embedding system into flat Riemannian or pseudo-Riemannian
3-manifolds $(N, \boldsymbol{h})$ is Darboux integrable. As an illustration of
the motivation behind the classification, we examine in detail one of the
classified metrics, $\boldsymbol{g}_0$, showing how to use its Darboux
integrability in order to construct all its embeddings in finite terms of
arbitrary functions. Additionally, the geometric Cauchy problem for the
embedding of $\boldsymbol{g}_0$ is shown to be reducible to a system of two
first-order ODEs for two unknown functions---or equivalently, to a single
second-order scalar ODE. For a large class of initial data, this reduction
permits explicit solvability of the geometric Cauchy problem for
$\boldsymbol{g}_0$ up to quadrature. The results described for
$\boldsymbol{g}_0$ also hold for any classified metric whose embedding system
is hyperbolic.
