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If I understand correctly, you're asking if a nonlinear system of PDE's is involutive, then is the corresponding linearized system also involutive? The answer is yes, but I'm not sure that it's obvious. It is true, as you say, that the symbol, which is the same for both systems, is involutive. However, involutivity also requires conditions on the lower order terms, and these need to be checked. I believe (but am not sure) that this is demonstrated in my monograph "Involutive Hyperbolic Differential Systems" for quasilinear systems.