An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive 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 


*

*$P$ is Lagrangian

*P involutive

*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.

 A: 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


*

*$\mathfrak p$ contains $\mathfrak g_x^{\mathbf C}$ and is $\mathrm{Ad}(G_x)$-stable;

*$\dim(\mathfrak p/\mathfrak g_x^{\mathbf C}) = \dim(\mathfrak g^{\mathbf C}/\mathfrak p)$;

*$\langle x,[\mathfrak p,\mathfrak p]\rangle=0$.
(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, 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.)
