Invariance of the Set of Extremal Solutions to a Conserved Quantity under Symmetries

Conserved quantities are functionals on solutions of an equation. The paper produces the explicit formula for the action of symmetries of the equation on these functionals and prove that the set of extremal solutions of any conserved quantity is invariant under this action. The Euler equation on extremals is not necessarily compatible with the underlying equation, but in the case of compatibility the extremals may present a rich source of exact solutions, as demonstrated by the example of the shallow water equation.