@MichaelBächtold Thanks for pointing out the "=" typo and the lack of specification for the $a^i_{[k],j}$. The infinite sum is necessary, because a priori it is not clear, up to which order one must go to get an integrable ideal. Also it is "not very likely" that for $N=100$ the Frobenius condition suddenly holds (for suitable chosen coefficieant functions), I am not aware of any condition to limit $N$. So that is the core of the question. However, some literature hints would also be welcome.

Another idea: in case of algebraic equations $h$ and $c_i$ Groebner Bases should provide a simple solution: Let $G$ be a Groebner Basis of the $c_i$ then $h \,\rm {mod} \,G$ should be $0$ (remainder should vanish). I tested a simple example with sympy, see docs.sympy.org/latest/modules/polys/…

that was just a typo. Thanks for pointing it out.