From one to infinity: symmetries of integrable systems

Abstract

Integrable systems constitute an essential part of modern physics. Traditionally, to approve a model is integrable one has to find its infinitely many symmetries or conserved quantities. In this letter, taking the well known Korteweg-de Vries and Boussinesq equations as examples, we show that it is enough to find only one nonlocal key-symmetry to guarantee the integrability. Starting from the nonlocal key-symmetry, recursion operator(s) and then infinitely many symmetries and Lax pairs can be successfully found.