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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.)

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A good introduction to integrable hierarchies is given by:

Miwa, Tetsuji, Michio Jimbo, and Etsuro Date. Solitons: Differential equations, symmetries and infinite dimensional algebras. Vol. 135. Cambridge University Press, 2000.

It is mostly focused on KdV and KP, so it is not complete, but it covers a number of important topics.

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It's not entirely clear what you mean by a "physical/geometric understanding of the corresponding integrable system" because each one will be unique in some sense.

A good collection of survey papers of the various techniques for analysing integrable systems is Y. Kosmann-Schwarzbach, B. Grammaticos and K.M. Tamizhmani, Integrability of Nonlinear Systems Lecture Notes in Physics 638, Springer 2004.

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As for the geometry, it indeed has many connections to integrable partial differential systems, see e.g. the book Geometry and Integrability surveying several aspects of the subject (twistors, geometry and self-dual Yang--Mills etc.), and e.g. this article for a connection to contact geometry.

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