In spite of such misgivings, the next system to receive attention was the harmonic oscillator. The constants of the motion of an isotropic harmonic oscillator are algebraically far simpler than those of a hydrogen atom, but nevertheless it seems that only the existence of the angular momentum had ever been suspected. Most likely the ease of solving the equations of motion of the harmonic oscillator, both classically and quantum mechanically, forestalled an active interest in finding the constants of the motion, whereas the Runge vector plays a very useful role in obtaining as well as exhibiting the solution of the hydrogenic problems. Nevertheless the awareness of the importance of knowing the symmetry group aroused by Fock's exposition of the symmetry of the hydrogen atom could hardly have failed to create interest in the symmetry of the harmonic oscillator, whose degeneracy was as well known as that of the hydrogen atom, and which is one of the most fundamental of all quantum mechanical systems.
The unitary unimodular group was found to be the symmetry group of the isotropic oscillator by Jauch (15) in 1939; a result which formed the principal content of his University of Minnesota doctoral dissertation (16), and which he and Hill published the following year in the Physical Review (17). A much more detailed treatment of the oscillator symmetry and related material including some historical and philosophical material on the development of quantum mechanics formed a very delightful set of seminar notes prepared by Hill (18), some years later, in 1954.
As had happened earlier with the hydrogen atom, the symmetry group of the oscillator was at first known only through the commutation rules of its constants of the motion. The geometric visualization which Fock's transformation had achieved was not immediately forthcoming, and in fact Jauch and Hill complained about the lack of apparent physical significance of some of their constants. The particular case of the two-dimensional harmonic oscillator was an interesting curiosity because it was definitely the unitary unimodular group and not its factor group, the three-dimensional orthogonal group, which was the symmetry group. This was a distinction which could readily be demonstrated because the two-dimensional oscillator has degenerate levels of every integer multiplicity, and only odd-dimensional representations can occur for the rotation group. Previously the group 
had always been associated with relativistic effects and spinning electrons, and not with purely classical problems.
It remained for Saénz, a student of Laporte, in his 1949 dissertation (19), at the University of Michigan, to exhibit a geometrically significant canonical transformation, whereby the phase space of the harmonic oscillator could be regarded as a complex vector space and the constants of the motion interpreted as generating unitary unimodular transformations of the phase space. The technique was to use the stereographic parameters of Laporte and Rainich, which in fact are the action-angle variables for the harmonic oscillator when properly parameterized.
The appearance of the harmonic oscillator Hamiltonian as a sum of squares of coordinates and momenta makes it very tempting to write it in a complex form, to which unitary transformations may be applied; but care must be taken to ensure that the unitary transformations are canonical, to preserve the spirit of Hamiltonian mechanics. Such an explanation of the unitary unimodular symmetry of the harmonic oscillator was published by Baker in 1956 (20).
The constants of the motion of the harmonic oscillator can be combined in complex form in such a way as to obtain the product of a creation and an annihilation operator; this interpretation also has a high intuitive significance, and suggests that other symmetry groups might be interpreted in terms of ladder operators.
At first sight, the analysis which applies to the isotropic harmonic oscillator ought to apply to the anisotropic oscillator as well, especially if one bears in mind the interpretation in terms of ladder operators and the exchange of quanta of energy between different coordinates. Jauch and Hill had found classical constants of the motion of the anisotropic two-dimensional oscillator, when there was a rational frequency ratio between the coordinates, but they found that the most reasonable quantum mechanical operator which they could construct based on the classical constants did not quite satisfy the proper commutation rules to define a Lie group. Dulock (21), in his University of Florida dissertation of 1964, managed to extend the classical constants to the general case of arbitrary frequency ratios, but it was even less clear how to form the quantum mechanical operators, in view of the transcendental nature of the constants.
Demkov (22), succeeded in avoiding the deficiencies in the commutation rules of the quantum mechanical ladder operators for the anisotropic oscillator -- in particular for the case of a 2:1 frequency ratio -- by dividing the states into two groups (of even and odd total energy, respectively) each one of which belonged to a unitary unimodular symmetry group. His student, Il'kaeva (23), treated a somewhat more general case, but as recently as 1968 Vendramin (24), published a claim that the unitary unimodular group cannot be the symmetry group of an anisotropic harmonic oscillator. Precisely stated, his claim is that the states of the anisotropic oscillator cannot belong to only one series of irreducible representations of the unitary unimodular group, which is in fact correct. Cisneros (25a) has unraveled the symmetry group of such systems, and has found that multiple families may occur, and additional degeneracies due to unitary groups of lower dimension when there are special relationships among the individual frequencies. That the representations of the unitary unimodular groups are reducible for rational frequency ratios was also analyzed by Maiella and Vilasi (25b).
Strictly speaking, the Jauch-Hill constants of the motion do not form a complete set of commuting constant for the anisotropic oscillator and one must additionally take into account some purely quantum mechanical operators. These operators determine the parity of the energy levels, or in the more general case, their residue classes with respect to the frequency of the individual oscillators. One has the interesting situation that the less commensurable the frequency ratios, the more families of representations of the unitary group occur, and the greater is the distance from the ground state to the first degenerate level. In the limiting case of incommensurability, the one-dimensional representation occurs infinitely often, and the gap to the first degenerate level is infinitely high, so to speak. In this way one can reconcile Dulock's result that even the incommensurable anisotropic oscillator has the symmetry group 
with the evident lack of any degeneracy, which would be untypical of a symmetry group.
The theme of Saénz' dissertation, which was aimed at an understanding of the symmetry of Dirac's equation for the hydrogen atom, was to reduce those systems wherein accidental degeneracy was known to occur to force-free motion on the surface of a hypersphere. The spherical top is another system which may be treated in such terms, although in that case the term ``accidental'' degeneracy is inaccurate because the hypersphere is the natural configuration space for the problem. We have already remarked that the rigid rotor was generally classified along with the harmonic oscillator and the hydrogen atom as a system showing accidental degeneracy, and for this reason it is interesting to find that there is a uniform treatment of all three systems.