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={{0,0,1,0},{0,0,0,1},{-1,0,0,0},{0,-1,0,0}} 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.