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In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...

First, why all the coefficients in the characteristic polynomial of L are elliptic functions, since the diagonal entries of the matrix L are the momentums? second, how to understand the ramification ...

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "...