Given a Kähler manifold $\mathcal{N}$ and an arbitrary (i.e. non-Kähler) differentiable sub manifold $\mathcal{M}\subset\mathcal{N}$, I would like to foliate $\mathcal{M}$ with maximal Kähler sub manifolds.
Intuitively, I can restrict the complex structure field $J$ from the Kähler manifold $\mathcal{N}$ using the orthogonal projection (using the Riemannian metric on $\mathcal{M}$). At every point $p$, there will be a maximal subspace $\overline{T_{p}\mathcal{M}}\subset T_{p}\mathcal{M}$ on which $J_{\mathrm{res}}^2=-1$. The choices of $\overline{T_{p}\mathcal{M}}$ forms a geometric distribution on $T\mathcal{M}$.
I would expect that I can typically (or always?) integrate the geometric distribution to find a collection of sub manifolds $\overline{\mathcal{M}}\subset\mathcal{M}$ that are all Kähler manifolds. I expect that this is standard and well-understood, but I did not find a discussion in the standard literature (which I'm not super familiar with).
It would be wonderful if somebody more familiar with the topic, could confirm (or correct) my suspicion and possibly refer me to a standard discussion.
I found the following reference: https://www.researchgate.net/publication/227124990_Differential_geometry_of_real_submanifolds_in_a_Kaehler_manifold.