Couldn't really fit the precise question in the title due to the character limit. I want to know what are some sufficient conditions for a model in classical field theory to possess infinitely many conserved quantities. The sine-Gordon and KdV equations are examples of such systems. Now, intuitely I believe that a system which allows Bäcklund transformations should fit the bill, for we can generate an infinite number of solitons. But I don't know if that's actually right and, if it is, I can't understand the reason as precisely as I would like. Is it? What are [other, if that is indeed one] sufficient conditions for a system to possess infinitely many conserved quantities?