I would like to know what is known (both explanations and references) about the spaces of smooth solutions to linear systems of PDEs of the following form:
Let $g_{1},...,g_{n}$ be smooth functions on $\mathbb{R}^{n}$ with the integrability condition $\partial{g_{i}}/\partial{x^{j}}=\partial{g_{j}}/\partial{x^{i}}$ and consider the space of smooth functions $f$ on $\mathbb{R}^{n}$ satisfying $\partial{f}/\partial{x^{i}}=fg_{i}$ for all $i$.
Similarly for the $g_{i}$ and $f$ being holomorphic on $\mathbb{C}^{n}$, and replacing $\mathbb{R}^{n}, \mathbb{C}^{n}$ with open contractible subsets.
My hope is that the answer is there is a unique solution, up to scaling.