My setting is the following: Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + \sum_{j=1}^{n-m} b^i_j\mathrm dF^j.$$
Roughly speaking, I am interested in the integrability of the Ideal $I_{[0]}:=\{\eta^1, ... \eta^m\}$.
'Normally', this is equivalent to the question: "Does the Frobenius condition $$ \mathrm d \eta^i\wedge \eta^1 \wedge ... \eta ^m =0 $$ hold for $i=1, ..., m$?"
However, in my case time derivatives are admitted: The coefficients $\omega^i = \sum_{j=1}^n \omega_j^i = \mathrm d x^j$ depend on the jet coordinates (without $t$): $\omega^i_j = \omega^i_j(x, \dot x, ...)$, where $x$ is itself an $n$-tuple. Additionally, $\mathrm d F^i$ has the form
$$\mathrm d F^i =\mathrm d f^i(x, \dot x) = \sum_{j=1}^n (f^i_j \mathrm d x^j + g_j^i \mathrm d \dot x^j),$$ where the rows $g^i$ of the matrix $(g_j^i)$ are linearly independent.
So, we extend the definition:
$$\eta^i_{[k]} := \sum_{j=1}^m a^i_{[k],j}\,L^k_{D_\chi}\omega^j + \sum_{j=1}^{n-m} b^i_{[k],j}\mathrm L^k_{D_\chi} dF^j.$$
Here, $L_{D_\chi}$ denotes the Lie-derivative with respect to the so called trivial Cartan vectorfiled: $$ D_\chi:= \frac{\partial}{\partial t} + \sum_{j=1}^n \dot{x}^j \frac{\partial}{\partial x^j} + ... \enspace . $$
Applying $L^k_{D_\chi}$ is thus equivalent to the total time derivative of order $k$, e.g. for $k=1$: $L_{D_\chi} (c \, \mathrm d x^1) = \dot c \mathrm d x^1 + c \mathrm d \dot x^1$.
Finally, we set $$ \eta^i_{<N>}:= \sum_{k = 0}^N \eta^i_{[k]}.$$
Now, I am interested in the (non)-integrability of the algebraic ideal $I = \{\eta^1_{<\infty>}, ..., \eta^n_{<\infty>}\}$. More specifically, given some additional information on $\omega$ and $f$, can one find an upper bound on $N$ above which additional time derivatives "do not help anymore" (to make the wedge product of the Frobenius condition vanish)? Most importantly: is it possible to state the integrability conditions of $I$ in terms of the given $\omega^i, \mathrm d F^i$ and its time derivatives?
Are there known results to problems like this? What would be the right terminology to search for. (I am aware of Books like those from Bryant et al., or Saunders. However, I would appreciate high-resolution-references like "Section x.y in Book ABC" or "Theorem of Foo-Bar"). I would also be grateful for corrections in my use of notation, concepts or terminology, as well for hints on improving the question.
