  We consider quantum motion on S3 perturbed by the trigonometric Scarf
potential (Scarf I) with one internal quantized dimensionless parameter, l, the
ordinary orbital angular momentum value, and another, continuous parameter, b,
through which an external scale is introduced. We argue that a loss of the
geometric hyper-spherical so(4) symmetry of the free motion occurs that leaves
intact the unperturbed hydrogen-like degeneracy patterns characterizing the
spectrum under discussion. The argument is based on the observation that the
expansions of the Scarf I wave functions for fixed l-values in the basis of
properly identified $so(4)$ representation functions are power series in the
perturbation parameter, b, in which carrier spaces of dimensionality (K+1)^2
with K varying as K\in [l, N-1], and N being the principal quantum number of
the Scarf I potential problem, contribute up to the order O(b^(N-1-K)).
Nonetheless, the degeneracy patterns can still be interpreted as a consequence
of an effective so(4) symmetry, i.e. a symmetry realized at the level of the
dynamic of the system, in so far as from the perspective of the eigenvalue
problem, the Scarf I results are equivalently obtained from a Hamiltonian with
matrix elements of polynomials in a properly identified so(4) Casimir operator.
The scheme applies to any dimension, d.
