An irreducible real analytic subvariety H of real dimension 2n+1\documentclass[12pt]{minimal}
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\begin{document}$$2n +1$$\end{document} in a complex manifold M is a Levi-flat subset if its regular part carries a complex foliation of dimension n. Locally, a germ of real analytic Levi-flat subset is contained in a germ of irreducible complex variety Hı\documentclass[12pt]{minimal}
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\begin{document}$$H^{\imath }$$\end{document} of dimension n+1\documentclass[12pt]{minimal}
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\begin{document}$$n+1$$\end{document}, called intrinsic complexification, which can be globalized to a neighborhood of H in M provided H is a coherent analytic subvariety. In this case, a singular holomorphic foliation F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} of dimension n in M that is tangent to H is also tangent to Hı\documentclass[12pt]{minimal}
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\begin{document}$$H^{\imath }$$\end{document}. In this paper, we prove integration results of local and global nature for the restriction to Hı\documentclass[12pt]{minimal}
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\begin{document}$$H^{\imath }$$\end{document} of a singular holomorphic foliation F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} tangent to a real analytic Levi-flat subset H. From a local viewpoint, if n=1\documentclass[12pt]{minimal}
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\begin{document}$$n=1$$\end{document} and Hı\documentclass[12pt]{minimal}
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\begin{document}$$H^{\imath }$$\end{document} has an isolated singularity, then F|Hı\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}|_{H^{\imath }}$$\end{document} has a meromorphic first integral. From a global perspective, when M=PN\documentclass[12pt]{minimal}
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\begin{document}$$M = \mathbb {P}^N$$\end{document} and H is coherent and of low codimension, Hı\documentclass[12pt]{minimal}
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\begin{document}$$H^{\imath }$$\end{document} extends to an algebraic variety. In this case, F|Hı\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}|_{H^{\imath }}$$\end{document} has a rational first integral provided infinitely many leaves of F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} in H are algebraic.
