I am learning about the Levi flat hypersurfaces. Let $M \subseteq \mathbb{C}^n$ be a real smooth hypersurface, i.e. for every point $p$ in $M$ there is an open set $U_p$ in $\mathbb{C}^n$ containing $p$ and a smooth function $r: U_p \to \mathbb{R}$ such that $grad(r)$ does not vanish on $U_p$ and $M \cap U_p=\{z \in U_p : r(z)=0\}$. Now, $M$ is said to be Levi flat if the Levi form of $M$ vanishes. Mathematically, the following holds: (Levi-flat condition) $\overset{2}{\underset{i, j=1}{\sum}}\bar{a}_ia_j\frac{\partial ^2 r}{\partial \bar{z}_i\partial z_j}|_p=0$ whenever $\overset{2}{\underset{i=1}{\sum}}a_i\frac{\partial r}{\partial z_i}|_p=0$ for all $p \in M$.
But I came across the following characterization for Levi-flat hypersurface which seems to be easier for checking:
Can someone explain what do we mean by “foliation” of $M$ by complex manifolds? I request to provide any suggestions/references for this topic.