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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.

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    You can rewrite your system as $\vec{\nabla} \ln f = \vec{g}$...2011-07-29
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    Does that help?2011-07-29
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    Also, what if $f$ has zeroes and what if I'm interested in complex values? I would be happy to say that there exists a contractible open cover of my space over which there is a unique solution on each covering set, if that would help solve the problem about branches of the logarithm.2011-07-29
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    Note that $f$ can only have zeros where $\vec{g}$ has singularities. Remember that $\vec{g}$ is the given and $f$ the unknown, so that when looking at the latter in terms of the former we are working with exponentials instead of logarithms and therefore don't have to worry about branches.2011-07-29

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