(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research level')
My setting is the following: Given two collections of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1,..., n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + \sum_{j=1}^{n-m} b^i_j\mathrm dF^j.$$
The symbols ($a_j^i$ and $b_j^i$) represent functions of $x = (x^1, ..., x^n)$. The matrix $(a_j^i)_{ji}$ is assumed to be regular everywhere.
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 $k$, $\frac{d^k}{dt^k}$. E.g. for $k=1$ we have: $L_{D_\chi} (c \, \mathrm d x^1) = \dot c \mathrm d x^1 + c \mathrm d \dot x^1$.
Finally, we set $$ \eta^i_{<N>}:= \sum_{k = 0}^N \eta^i_{[k]}.$$
[Edit1] The coefficient functions $b_{[k],j}^i$ are arbitrary, while the functions $a_{[k],j}^i$ must fulfill the following condition: The matrix $$ A:= \sum_{k = 0}^N (a_{[k],j}^i)_{ji} \frac{d^k}{dt^k} \tag{*} $$ which is a polynomial matrix of the time derivative operator $\frac{d}{dt}$ (=$L_{D_\chi}$) must be "unimodular", i.e. it must have an inverse which is also a polynomial matrix of $\frac{d}{dt}$. Unfortunately, unlike the case of constant coefficients, there is no determinant-criterion for unimodularity applicable here.[/Edit1]
Now, I am interested in the (non)-integrability of the algebraic ideal $I = \{\eta^1_{<\infty>}, ..., \eta^n_{<\infty>}\}$.
[Edit3] In other words: Are there functions $h^i = h^i(x, \dot x , ...)$ such that $\mathrm d h^i = \eta^i_{<N>}$ for some fixed $N\in \mathbb N$ and $i = 1, ..., m$. ($a$ and $b$ are arbitrary up to condition (*)) If such functions $h^i$ do not exist: How can this be shown? [/Edit3]
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?
[Edit2] The background of this question lays in control theory, in particular in the theory of differential flatness. $\mathrm d F^j$ comes from the system under consideration and $\omega^i$ is the result of an special algorithm for determining a dual basis of the 'tangent system'. If $\eta$ is integrable, then, roughly speaking, $h$ with $\mathrm d h = \eta$ is the desired flat output mapping. [/Edit2]
