This looks interesting. I will try to implement and test it an then report it here.

Thank you. I corrected the question.

I added this to the question: $N$ can be choosen arbitrarily. In general $\mathbf q$ contains $2(N+1)$ free parameters and the $n$ given points together pose $2n$ conditions. Thus I would suppose that $(N+1) \geq 2n$ is a condition for solvability. But there might be additional constraints come into play to ensure that the curve is closed (may be better formulated: has at least one closed branch).

@Igor You are right. The question boils down to the existence of such functions $h^i$. However I do not understand what you mean with the condition $\omega^i=const$ and $F^i=const$. The forms $\eta^i_{<N>}$ are linear combinations of $\omega^i$, $\mathrm d F^i$ and their time derivatives (due to condition (*) in such a way that this transformation can be inverted w.r.t the $\omega$^i). IMHO this completely defines the relation between $\eta$ and $\omega, \mathrm d F$. Further, I do not even understand what $\omega = const$ means. (Maybe on the level of coefficients like e.g. $3dx_1 + 5dx_2$?)

I added some background information but I fear to provide a substantial motivation I would have to write a lot more. Therefore I tried to state the problem purely mathematical, without referring on motivation. BTW: apart from the unimodularity condition the $a_{[k]_j^i}$ are arbitrary ($b_{[k]_j^i}$ with out restriction). The question is: Do they exist, such that $\eta_{<N>}$ is integrable. Or: how to show that they do not exist.