[foxes]

[0] Yes a solution will transform under the symmetry to be another solution of the same equations. Perhaps for a bit of history Galois originally used finite groups to look for solutions to algebraic equations. You might find one solution and then be able to describe all the other solutions. A bit later on Sophus Lie (where the name Lie groups comes from) went on to study the same problem for differential equations. Maybe to say it briefly, if you understand that your solutions have a symmetry you can use it to simplify your original equation and make things easier to solve.

In the field theory example, you have something called a Lagrangian which you can use to derive the equations which describe the dynamics of your system. In those cases it will have a symmetry of one of those groups, you can transform the Lagrangian by one of those groups.

Knowledge of the symmetry does help you simplify your problem - its basically used to remove redundancy.