Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^k$ and consider the system of $k$ equations $$ \sum_{i=1,\dots,n; j=1,\dots,k} \xi^{(m)}_{i,j} \frac{\partial f_i}{\partial x_k} = f_m, $$ with $m$ ranging from $1$ to $k$, where $\xi^{(m)}_{i,j} : \mathbb{R}^n \rightarrow \mathbb{R}$ are some functions.
Is there a general theory for the solution of such systems of equations?
As a matter of fact, my case is particularly easy: here we have $\xi^{(m)}_{i,j} \in \{-1,0,1\}$. In fact, so many of them are zero that it is not difficult to guess a family of solutions. But I would like to find all possible solutions.
In my case I also have $n=k=6$, so the problem is not extraordinarily high-dimensional. I am also happy if someone can suggest computational solution that will give me a family of approximated solutions.
The general theory for the solution of such PDEs is the theory of Exterior Differential Systems. The method of Cartan-Kähler Calculus works in the real-analytic category, which was enough for my purposes.
The monograph Bryant, R.L., Chern, S.S., Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A.: Exterior Differential Systems gives an introduction to the theory. An example in the spirit of my question is executed on pages 60-63.
The DifferentialGeometry package of the mathematical software Maple has these exterior differential systems implemented, and my problem can be solved using the IntegralManifold method. In fact, even the standard method pdsolve is powerful enough to solve such systems.