Noetherian first integrals

From time to time one finds claims in the literature that first integrals/invariants of Lagrangian systems are nonnoetherian. Such claims diminish the contribution of Noether in the topic of integrability. We provide an explicit demonstration of noetherian symmetries associated with the integrals which have been termed nonnoetherian. To further emphasise our point we construct the noetherian first integrals/invariants, which are associated with symmetries linear in the velocities, for the two-dimensional autonomous isotropic harmonic oscillator and the autonomous anisotropic oscillator and illustrate the roles which the invariants can play in the description of the classical motion. We relate these symmetries to the corresponding problem in quantum mechanics. Further we show that the complete symmetry group of this anisotropic harmonic oscillator has the same representation as that of the corresponding isotropic oscillator. As a concluding example we show that a symmetry claimed to be nonnoetherian is trivially Noetherian.