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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.

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

EDIT: I see that there's also a decent write-up on Wikipedia, you just need to know where to look.