0 Introduction

What is an integrable system? Unfortunately, an integrable system is a some-

thing mathematicians have not yet managed to define properly. Intuitively, an

integrable system is a differential equation we can “integrate up” directly. While

in theory, integrable systems should be very rare, it happens that in nature, a

lot of systems happen to be integrable. By exploiting the fact that they are

integrable, we can solve them much more easily.