The application of Noether's theorem in this case refers only to the energy integral shown (KE = ME - GPE for 2D Kinetic Mechanical and Gravitational Potential Energies) over time. It's really only for that particular 2 body 2 dimensional problem.
More generically in 3 dimensions a transformation with 3 translational 2 rotational and 1 time independence would provide conservation of 3 momenta 2 angular momenta and 1 energy.
More generically in 3 dimensions a transformation with 3 translational 2 rotational and 1 time independence would provide conservation of 3 momenta 2 angular momenta and 1 energy.