Let $(M,\omega)$, be a symplectic manifolds. 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. We know that polarized sections are covariantly constant along the leaves of $D$,.

I am looking for an example to show that when these leaves be non-compact then these polarized sections are not square integrable with respect to volume form $\omega^n$.