Fix a vector space $V$ and an integer $1\leq n<\dim V$.
If $\mathcal{I}\subseteq\Lambda^\bullet V^*$ is an ideal, I denote by $\mathcal{I}^i:=\mathcal{I}\cap\Lambda^iV^*$ its $i^\textrm{th}$ homogeneous piece, and by $$ \ker\mathcal{I}^i:=\bigcap_{\omega\in \mathcal{I}^i}\ker\omega\subset\Lambda^iV $$ the linear subspace of $\Lambda^iV$ made by the elements which vanish when contracted with all the $i$-forms from $\mathcal{I}$.
Denote by $$ K(\mathcal{I}):=\prod_{i=1}^n\mathbb{P}(\ker \mathcal{I}^i)\subset\prod_{i=1}^n\mathbb{P}(\Lambda^iV)\subset\mathbb{P}\left( \bigotimes_{i=1}^n\Lambda^iV\right) $$ the Segre-image of the product of the projectivised kernels above (the reason why I start from $i=1$ is that I'm assuming $\mathcal{I}^0=0$).
Finally, I need the flag variety $$ \mathrm{Fl}(1,\ldots,n,V)\subset\mathbb{P}\left( \bigotimes_{i=1}^n\Lambda^iV\right) $$ the embedding being given by $$ (\langle v_1\rangle,\langle v_1, v_2\rangle,\ldots,\langle v_1,v_2,\ldots,v_n\rangle)\mapsto [v_1\otimes(v_1\wedge v_2)\otimes\cdots\otimes(v_1\wedge v_2\wedge\cdots\wedge v_n)]\, . $$ These two ingredients allow me to define the flags of integral elements $$ \mathrm{Fl}_n(\mathcal{I}):=K(\mathcal{I})\cap\mathrm{Fl}(1,\ldots,n,V)\, . $$
Motivation. In the theory of Exterior Differential Systems, if $V=T_xM$, then an element $(V_1,V_2,\ldots, V_n)$ of $\mathrm{Fl}_n(\mathcal{I})$ is called an integral flag at $x\in M$. If such a flag is a ``good one'', then one can iteratively use the Cauchy-Kowalewskaya theorem to construct the germ at $x$ of an integral submanifold of $V_n$.
Define now a sub-variety $$ \mathrm{Fl}^{\textrm{ord}}_n(\mathcal{I})\subseteq\mathrm{Fl}_n(\mathcal{I})\, , $$ by requiring it to be the largest one on which all the natural bundle $$ \mathrm{Fl}(1,\ldots,n,V)\longrightarrow \mathrm{Gr}(i,n)\, ,\quad i=1,\ldots,n\, ,\quad (^*) $$ restrict to sub-bundles (by sub-bundle I mean that both the fibre and the base can get smaller).
Motivation. The "good flags" above are called "ordinary" in the Cartan-Khaler theorem (the ranks of the restricted bundles corresponding to the Cartan characters). So, to be able to spot ordinary flag is to be able to integrate an EDS.
After this lengthy intro, I can finally formulate my
QUESTION: is there some algebraic machinery to construct $\mathrm{Fl}^\textrm{ord}(\mathcal{I})$ out of $\mathrm{Fl}(\mathcal{I})$? In particular, is there a test to recognise when the two are equal?
Some comments. Here's what I would expect: maybe there is some sort of "completion" $\mathcal{I}^\textrm{ord}\supseteq \mathcal{I}$, such that $$ \mathrm{Fl}^\textrm{ord}(\mathcal{I})=\mathrm{Fl}(\mathcal{I}^\textrm{ord})\, , $$ and I expect that $\mathcal{I}^\textrm{ord}$ can be costructed in a purely algebraic manner out of $\mathcal{I}$. Or, in a similar perspective, maybe there is a natural way to single out a "core" $K^\textrm{ord}(\mathcal{I})\subseteq K(\mathcal{I})$, such that $$ \mathrm{Fl}^\textrm{ord}(\mathcal{I})=K^\textrm{ord}(\mathcal{I})\cap\mathrm{Fl}(1,\ldots,n,V)\, . $$ In both cases, I have not even the slightest idea of how to construct these guys. I simply suspect that they exist.
The reason which led me to believe so, is that the "multi-bundled structure" of $\mathrm{Fl}(1,\ldots,n,V)$ is captured by the natural completely integrable distributions $\Delta_i$ it is equipped with (the maximal integral submanifolds of $\Delta_i$ being precisely the fibres of $(^*)$, for $i=1,\ldots,n$). So, in principle, it all boils down to study the integrability of the restricted distributions $\Delta_i|_{\mathrm{Fl}(\mathcal{I})}$, but I'm not able to go further.
Any reference to similar problems already dealt with in the literature, will be warmly appreciated!
