There are lots of papers on say, W-algebras, that relate them to integrable systems like KdV, the KP hierarchy, etc. Algebraically this is done just by writing down infinitely many commuting operating, which is very pretty but leaves me without a physical/geometric understanding of the corresponding integrable system.

Which brings me to my question: What's a good reference to learn about the importance of and classical approach to these integrable systems? (KdV, Toda, KP...)

(Some references appear in the answers to this question, but most of them concern finite-dimensional integrable systems. It seemed to me that my question merited a separate thread.)