I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which decouples from the rest, determining $4$ dependent variables, which written in matrix notation is

$u_{t,i} + B_{i j}u_{x,j}+ C_i=0$.

Here $$B=\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix}$$ so has doubly degenerate eigenvalues $\pm i$. I gather that because these eigenvalues are complex, this is an elliptic system of first-order linear PDE's. Can anybody recommend a good reference which would help me see/explain to me whether and how this elliptic system could be solved analytically?

So far I have come across few references which go into detail about how to solve a system of first-order linear PDE's, in particular ones that are not totally hyperbolic.