I consider a system of partial differential equations in the region $V \subset R^n$, given by $\frac{\partial u}{\partial x_i} = F_i(x_1, \cdots, x_n)$.
Each $F_i$ is assumed to be infinitely differentiable with respect to each $x_i$ in $V$.
The initial condition is set as $u(0, \cdots, 0) = a$ (constant), where $(0, \cdots, 0) \in V$.
In this case, can we ensure the existence of a unique local solution in a sufficiently small neighborhood arround $(0, \cdots, 0)$ ?
Since each of the $F_i$ terms is independent of $u$, your system can be rewritten as a gradient equation: $$ \nabla u = F = (F_1,\dotsc,F_n). $$ If there were a $C^2$ solution, then by the commutativity of mixed partials we would have to have $$ \partial_i \partial_j u = \partial_j \partial_i u \text{ of equivalently } \partial_i F_j = \partial_j F_i. $$ If you just want a $C^1$ solution, then there is still a version of this that must hold in the sense of distributions. So, you see from this that there is an obstruction to solving your system for arbitrary $F$. However, if the $F_i$ satisfy this condition, then you can use one of the versions of the Frobenius theorem (probably the infinite dimensional formulation on wikipedia is the clearest connection to what you want, but it's significantly more general) to get your local solution $u$.
