# How to use these higher symmetries and conservation laws?

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation.

However, unlike the classical symmetries (point symmetries), the higher symmetries (or Lie–Bäcklund symmetries; such as KdV hierarchy) seem useless, or there are something I am unfamilar.

Similar case are the conservation laws. For KdV equation, we have infinitely many $$\int u\,dx, \int u^2dx, \int \frac12u_x^2-u^3dx, \dotsc$$. But it seems that only the first few conservation laws are useful.

I know some people treat the existence of infinite symmetries or conservation laws as a criterion for whether the equation is integrable, but I don't see the real application.

The question is: how to utilize these infinitely many symmetries and conservation laws?

It should be stressed that the other way around this is not true: there is plenty of integrable systems, especially in the case of more than two independent variables, which possess Lax pairs and other attributes of integrability but have no genuinely higher symmetries, although they often have infinitely many nonlocal symmetries (and nonlocal conservation laws), cf. e.g. Sergyeyev - A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems, Sergyeyev - A simple construction of recursion operators for multidimensional dispersionless integrable systems, Sergyeyev - New integrable (3+1)-dimensional systems and contact geometry (published version), and Jing Ping Wang - On the structure of $$(2+1)$$-dimensional commutative and noncommutative integrable equations, and references therein.