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-Backlund 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 infinite many $\int udx$, $\int u^2dx$, $\int \frac12u_x^2-u^3dx$, $\cdots$. 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 whether the equation is integrable, but I don't see the real application.

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