  We analyze two conditionally solvable quantum-mechanical models: a
one-dimensional sextic oscillator and a perturbed Coulomb problem. Both lead to
a three-term recurrence relation for the expansion coefficients. We show
diagrams of the distribution of their exact eigenvalues with the addition of
accurate ones from variational calculations. We discuss the symmetry of such
distributions. We also comment on the wrong interpretation of the exact
eigenvalues and eigenfunctions by some researchers that has led to the
prediction of allowed cyclotron frequencies and field intensities.
