I know that once we find the Lax pair $\dot{L}=[L,M]$ we could get the conserve quantities $H_k = \frac{1}{k}{\rm Tr}L^k$. But it seems that there is not general way to construct the Lax pair (e.g. in How to find a lax pair of an integrable system given the PDE or Hamiltonian? ) someone commented that
in practice, there are no general methods of computing a Lax pair for a given system of PDEs
It seems that Lax pair could only be obtained case by case. So I wonder why it lies in the center of the study of integrable system.
- Does Lax pairs exist (though may be hard to construct) for any integrable system?
- What could Lax pairs tell us about the generic feature of integrable systems, e.g. the classification of integrable systems?