I modified the question because it was confused.
On my book there is this mathematical definition of symmetry transformation:
"The equations of motion have a symmetry, if the solutions of the equations transformed by the symmetry are still solutions of the equations of motion, namely, there is a symmetry if the transformed equations of motions have the same form of the original".
I don't understand the meaning of this sentence, do you think is it a good (and easy) mathematical definition of symmetry transformation? Anyway what "equal in form" means? Then, i know rotation of an isolated system is a symmetry, can you make an easy example of an isolated system in which if we apply a rotation the mathematical definition of symmetry apply? Can you make an example of non symmetric transformation and show me why the mathematical definition doesn't apply?