Every maximal isotropic (totally null) subspaces $L\subset V\oplus V^*$ (with respect to the natural pairing $\langle X+\xi,Y+\eta\rangle = \eta(X) + \xi(Y)$) can be express as $L(E,\alpha)$ for some appropriate $E⊆V$ and 2-form $α∈Λ^2(E)$:
$$ L(E,\alpha) = \{X+\xi\in E\oplus V^* : \xi|_E = i_X\alpha \} . \tag{1} $$
I already asked a question about the proof of this fact here.
Now, I'm wondering if every maximal isotropic subbundle $L\subset TM\oplus T^*M$ can also be expressed in the same way. Namely, if given $L$, there exists a subbundle $E\subset TM$ and 2-form $\alpha\in\Gamma^\infty(\Lambda^2E)$ such that
$$L=\{(p,X)+(p,\xi)\in E\oplus T^*M : \mbox{for each } p\in M \;\; \xi|_{E_p} = i_X \alpha \} . $$
According to Gualtieri, this identification is possible only in regular points $p\in M$, in the sense that the leaf dimension is constant in a neighbourhood $U$ of $p$. But I can't figure why leaves are important here. To prove my claim I thought of using the vector bundle construction theorem using as typical fibe the vector space of (1), where in this case $E$ would be the typical fibre of the vector subbundle $E$.
What do you think?
Context. This question arises in the proof of the Generalized Darboux theorem (Gualtieri's thesis pp.56--57). To prove the theorem he uses a previous proposition 3.12 about transverse folitations. Then, he must to restrict the Darboux theorem to regular points of the manifold. However, I think that a maximal isotropic subbundle can be expressed in the above form, independently the regularity of the point of the manifold but he asys We saw in Proposition 4.19 that in a regular neighbourhood, a generalized complex structure may be expressed as $L(E, ε)$ where $E < T ⊗ C$ is an involutive subbundle and $ε ∈ C^∞(∧^2E^∗)$ satisfies $d_Eε = 0$. Probably he is right but why do we need regularity?
PD: Feel free to add more tags. I have added only one because there isn't any about generalized geometry.