System of linear first order homogeneous PDEs

Definition
A system of first order linear homogeneous PDEs can be written in the general matrix form as

A_{0} u_{t} + A_{1} u_{x} + A_{2} u_{y} + A_{3} u_{z} = F,

where $x, y, z, t$ are independent variables, $u = (u_{1}, \dots, u_{N})^{T}$ is an $N$ -dimensional vector of unknown functions, and $A_{i}$ are $M \times N$ matrices whose entries depend on the independent variables $x, y, z, t$ . $F$ is a vector also depending on $x, y, z, t$ . Both $M$ and $N$ are positive integers satisfying $M \geq N$ .
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Existence of solutions

See this answer of R. Bryant