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?