I would be interested in a collection of existence and uniqueness theorems for systems of linear first-order partial differential equations. The books on PDEs that I have looked through either do not cover systems of PDEs or only mention theorems requiring one assumption or another on the functions involved in the equations. So I was wondering what the current state of the art is regarding general solutions to such systems.
Let $\mathbf{u}: U \rightarrow \mathbb{R}^m$ be a vector-valued unknown function, $\mathbf{u} = (u^1(\mathbf{x}), \ldots, u^m(\mathbf{x}))$, of $n$ independent variables $x_j$, so $\mathbf{x} = (x_1, \ldots, x_n)$. The domain $U \subseteq \mathbb{R}^n$ is open, possibly with boundary $\partial U$. Then a system of linear first-order partial differential equations can be written $$ \sum_{r=1}^m A_r(\mathbf{x}) \, \frac{\partial \mathbf{u}(\mathbf{x})}{\partial x_r} + B(\mathbf{x}) \, \mathbf{u}(\mathbf{x}) = \mathbf{f}(\mathbf{x}) $$ for given real-valued functions $\mathbf{f} = (f^1(\mathbf{x}), \ldots, f^m(\mathbf{x}))$. Here, $A_r$ and $B$ are matrices with elements $a_{ij}^r(\mathbf{x})$ and $b_{ij}(\mathbf{x})$ that are real-valued functions of the independent variables.
In particular, I am interested in the following questions: for a1) $U$ a subregion of $\mathbb{R}^n$ with boundary $\partial U$ and a2) $U$ all of $\mathbb{R}^n$, what existence and uniqueness theorems can be formulated when b1) $A_r$, $B$ and $\mathbf{f}$ are as general as possible (fewest assumptions such as being continous, analytic,... imposed on them). More specifically, what about the cases where b2) the given functions $a_{ij}^r$, $b_{ij}$ and $f^j$ are integrable on the domain $U$, b3) the elements $a_{ij}^r$ and $b_{ij}$ are constants. Additionally, b4) are there theorems for when any of $A_r$, $B$ and $\mathbf{f}$ are not functions, but distributions?
I don't necessarily expect answers to address all cases listed above, but I could compile such a list based on individual answers. Any answers should contain references to books or articles with a clear exposition of the theorem relevant for each case.