When the RHS is 0, you are basically asking about the method of Riemann invariants. A quick summary of the method you can find in the second section of 

[Lax (1964), J. Math. Phys.](https://mathscinet.ams.org/mathscinet-getitem?mr=165243)

Alternatively, I am pretty sure it is also discussed somewhere in the second volume of Courant and Hilbert.