The state of affairs as Saénz presented it in 1949 was reasonably complete. The three most widely known systems in which accidental degeneracy above and beyond that required by spherical symmetry was known to exist were shown either by an appropriate choice of coordinates or an appropriate canonical transformation to be equivalent to force free motion of a particle confined to the surface of a hypersphere (or in the case of the positive or zero energy states of the hydrogen atom, a closely related surface). Some slight extensions of this result in other directions was known. For example, although a physicist would naturally give his predominant attention to three-dimensional systems such as occur in the real world, some two-dimensional systems were considered important enough to treat, and sometimes results were stated generally for an arbitrary number of dimensions. For example, Jauch and Hill had shown that a transformation similar to Fock's would account for the two-dimensional hydrogen atom, and Saénz' results were stated in quite general terms even if they were only applied to the familiar low-dimensional systems. In 1958 Alliluev (46) extended the stereographic technique explicitly to the hydrogen atom in spaces of an arbitrary dimensionality, and Loudon (47), in 1959 found an unexpected degeneracy for the one-dimersional hydrogen atom, although this conclusion rests on some delicate assumptions concerning the appropriate boundary conditions for the Schrödinger equation which are not as yet understood to everyone's satisfaction.
Arguments existed to the effect that these were the only systems which ought to have exhibited accidental degeneracy, although there were sufficient loopholes in the argument to admit a number of other known highly degenerate systems. For example, the Kepler problem in non-Euclidean spaces was known to behave much like its Euclidean version, although the discrete spectrum was finite in a hyperbolic space (48), and the spectrum of the unbound states was discrete in spherical space (49). Even a free particle enclosed in an impenetrable cubic box also showed more degeneracy than cubic symmetry would require (50).
The usual argument supporting the uniqueness of the harmonic oscillator and Keplerian potentials in forming accidentally degenerate systems is an appeal to Bertrand's (51) theorem enunciated in 1873, which states that in problems where the kinetic energy is the sum of the squares of the velocities (which exempts non-Euclidean spaces and vector potentials) and the potential is spherically symmetrical (which exempts the cubical square well and anisotropic harmonic oscillator), the only potentials with bounded closed orbits are those for 
(Coulomb potential) and
(harmonic oscillator). As we have remarked, the degeneracy of the tops is not technically ``accidental.'' The reason for the strict limitations in Bertrand's theorem are of course due to its method of proof, since one separates the Hamilton-Jacobi equation with the assumed potential and kinetic energies, assumes the existence of a circular orbit, and makes a perturbation calculation to see whether orbits near the circular one are also closed.
Bertrand's original paper is nowadays rather inaccessible, but fortunately there is an excellent exposition of the theorem and discussion of accidental degeneracy in a recent article of Greenberg (52). The same lines of reasoning could well enough be applied to systems with different forms of kinetic energy. Such an analysis was actually made by Darboux (53) for motion confined to a surface of revolution, a few years after Bertrand's theorem was published. He likewise investigated the possible force laws which could produce elliptic orbits (54a). An extension of Bertrand's theorem which is of interest for the movement associated with a magnetic monopole and some similar fields was published by Lehti (54b), in 1968.
Implicit in an application of Bertrand's theorem to limit the number of accidentally degenerate mechanical systems is the assumption that accidental degeneracy is to be equated with the existence of bounded, closed orbits; or in any event with orbits which are not space filling. There is also an implicit assumption that the existence of such orbits in the classical problem will be a reliable indicator of the degeneracy in the quantum mechanical problem.
Since the mathematical understanding of these conditions has never been completely decisive, there has grown up a considerable folklore about the nature of accidental degeneracy, reflecting to a considerable extent people's aesthetic feelings about how the resolution ought to appear. Mainly, we know of certain conditions under which accidental degeneracy will arise, but it is largely a matter of faith (and our limited range of experience) that there will be no accidental degeneracy when these conditions are absent. More bluntly put, we know some sufficient conditions for accidental degeneracy, but we do not as yet know any necessary conditions expressible in intuitive physical terms. One of the more attractive tenets of the folklore has been that accidental degeneracy arises from hidden symmetry, as has been amply demonstrated by the major examples. Hidden symmetry of course implies a symmetry group, from which degeneracy follows in a pattern dependent upon the occurrence and dimensionalities of the irreducible representations of the group of the hidden symmetries.
Assume, as most often seems to be the case, that the symmetry group is a continuous Lie group. It will have a certain number of infinitesimal generators, which will necessarily be constants of the motion. Quantum mechanically, these generators will transform eigenfunctions of the Hamiltonian into other eigenfunctions, and so the possibility for degeneracy exists, unless the Hamiltonian and the constants of the motion have exactly the same eigenvectors. This is not possible when there are several noncommuting constants of the motion, say when the symmetry group itself is non-Abelian. Classically, the constants of the motion must generate transformations of orbits in manifolds of constant energy so that the more independent constants which exist, the more automorphisms of the constant energy manifolds we will have to work with.
For the case of a symmetry group, these remarks are but an informal statement of Schur's lemmas and the mechanism by which symmetry results in degeneracy. However, the existence of an adequate number of independent constants of the motion from whatever source they may arise will also produce degeneracy. One such source might be the separation of Schrödinger's equation (or alternatively the Hamilton-Jacobi equation) in several distinct coordinate systems. Separability in several coordinate systems is known to be related to the existence of orbits that are not space filling, as well as to classical degeneracy.
The reasoning is the following: If the Hamilton-Jacobi equation is separated, and action-angle variables exist, the existence of distinct frequences for the angle variables will result in a Lissajous figure type of motion which will fill a region of phase space if the frequencies are incommensurable. The boundaries of the region will be surfaces where one or another coordinate takes an extreme value, so that the coordinate surfaces will be inherently defined by the orbits. For the bounding surfaces to coincide for two distinct systems of coordinates, the frequences must be commensurable if not actually equal, which is just the meaning of classical degeneracy. Since separability of the Schrödinger equation goes hand in hand with separability of the Hamilton-Jacobi equation the reasoning may be extended from the classical to the quantum mechanical version of a problem. We may therefore always suspect a classically degenerate problem of originating a quantum mechanically degenerate version.
Degeneracy in a classical problem may also tell us something about its constants of the motion, as to whether they are algebraic functions of the coordinates and momenta or not. For, if a problem has a complete set of algebraic constants, one may define the coordinates and momenta in terms of them. For the moment we would want to define one coordinate in terms of another and some of these algebraic constraintss, which would imply that one coordinate was a finite-valued function of other, and hence the orbit could not be space filling. We can therefore infer that when a system has a complete set of algebraic constants of the motion, it would be classically and thereby quantum mechanically degenerate. Few systems satisfy this requirement; an important theorem demonstrated by Poincare at the beginning of the century denied such a possibility for the three-body problem in celestial mechanics, for example.
Summarizing the foregoing analysis, we can reach several tentative conclusions.
1. Symmetry, hidden or open, produces a degeneracy, to a degree depending on the noncommutability of the generators of the symmetry group.
2. Constants of the motion do the same. Since the Poisson bracket (or commutator) of two constants will be another constant of the motion, they will generate a Lie group whenever the set is so closed, but they will induce degeneracy regardless.
3. For a system to be separable in more than one coordinate system, orbits must not be space-filling. In this case there will be an abundance of separation constants.
4. When there is a complete set of algebraic constants, the orbits will not be space filling. Not many problems are known with such a set, and some are known not to be so endowed.
5. When the kinetic energy has its usual form and the potential is spherically symmetric, the only potentials for which no orbit is space filling are the Kepler and harmonic oscillator potentials. These, by explicit construction, are known to have, respectively, 
and 
as symmetry group accounting for their accidental degeneracy.
Basing one's conclusions on the above analysis, it is seen that Bertrand's theorem does not state that the known degenerate systems are the only ones, but it does ensure that other systems will either have space filling orbits or violate such assumptions as spherical symmetry of the potential or the Euclidean form of the kinetic energy. Since problems with space-filling orbits are not classically degenerate, it seems somewhat unlikely that they would be found to be quantum mechanically degenerate.
One precaution which must be observed in reviewing these conclusions is that the considerations have not always been stated with full mathematical precision and the results deduced and stated rigorously. This will be seen to lead to some further loopholes. Nevertheless, we have a fair representation of the situation as it existed until very recently, and as expounded in several sources— Saénz (19), Greenberg (52), or Whittaker (55), for example.