I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if 


1. $P$ is Lagrangian
2. P involutive
3. dim$P\cap\bar P \cap TM$ is constant


This definition shows that every complex polarization induces a real isotrpic distribution $D:=P\cap\bar P \cap TM$ which is also involutive by Frobenius theorem. Moreover the complexification of distribution $D$ is $D^{\mathcal{C}}=P\cap \bar P$ and it is called isotropic distribution. Now we define the subbundle $E:=(P+\bar P)\cap TM$ and $E^{\mathcal{C}}=P+ \bar P$. Notice that orthogonal symplectic complement of $D$ is $E$, i.e., $D^{\perp}=E$ and $E$ is called coisotrpic distribution.



> 
> I am looking for an example to show that subbundle $P+ \bar P$ is not
>  necessarly involutive in general.
> 
> 
> 



An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive
Suppose your $M$ is the coadjoint orbit of a Lie group $G$ through $x\in\mathfrak g^\*$; write $G\_x$ for the stabilizer of $x$. The set of $G$-invariant polarizations $P$ on $M$ identifies with the set of subalgebras $\mathfrak p\subset \mathfrak g^{\mathbf C}$ such that


1. $\mathfrak p$ contains $\mathfrak g\_x^{\mathbf C}$ and is $\mathrm{Ad}(G\_x)$-stable;
2. $\dim(\mathfrak p/\mathfrak g\_x^{\mathbf C}) = \dim(\mathfrak g^{\mathbf C}/\mathfrak p)$;
3. $\langle x,[\mathfrak p,\mathfrak p]\rangle=0$.


(Just send $P$ to the preimage $\mathfrak p$ of $P\_x\subset T\_xM^{\mathbf C}$ under the infinitesimal action $\mathfrak g^{\mathbf C}\to T\_xM^{\mathbf C}$.) In this setting, your condition that the distribution $P+\bar P$ be involutive translates into the condition that $\mathfrak p + \bar{\mathfrak p}$ be a *subalgebra* of $\mathfrak g^{\mathbf C}$.


It is not hard to cook up examples (with $G$ semisimple) failing it. In particular, in Example 6.1 of [On polarizations of certain homogenous spaces](http://www.ams.org/mathscinet-getitem?mr=340483), Ozeki and Wakimoto construct an $e\in\mathfrak{su}(3,3)^\*$ whose orbit only has polarizations such that $\mathfrak p + \bar{\mathfrak p}$ is not a subalgebra. (More precisely it has "w-polarizations", failing the $\mathrm{Ad}(G\_x)$-stability condition in 1.; as explained on p. 447 of the paper these correspond to bona fide polarizations $P$ of the orbit's universal covering.)


