Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold (M,g)\documentclass[12pt]{minimal}
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\begin{document}$$(M, \varvec{g})$$\end{document} and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold (N,h)\documentclass[12pt]{minimal}
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\begin{document}$$(N, \varvec{h})$$\end{document}, one can ask under what circumstances does the exterior differential system I\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {I}$$\end{document} for an isometric embedding M↪N\documentclass[12pt]{minimal}
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\begin{document}$$M\hookrightarrow N$$\end{document} have particularly nice solvability properties. In this paper we give a classification of all 2-dimensional metrics g\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{g}$$\end{document} whose isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds (N,h)\documentclass[12pt]{minimal}
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\begin{document}$$(N, \varvec{h})$$\end{document} is Darboux integrable. As an illustration of the motivation behind the classification, we examine in detail one of the classified metrics, g0\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{g}_0$$\end{document}, 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 g0\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{g}_0$$\end{document} 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 g0\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{g}_0$$\end{document} up to quadrature. The results described for g0\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{g}_0$$\end{document} also hold for any classified metric whose embedding system is hyperbolic.
