The equation I'm interested in is (1-u2)P''(u) - 2uP'(u) + (λ - βu - m2/(1-u2))P(u) = 0, where m takes on integer values, λ is an eigenvalue, and β is a known constant. Also, there is a change of basis where u = cosθ. It is similar to the Legendre's associated equation (1-u2)P''(u) - 2uP'(u) + (λ - m2/(1-u2)P(u) = 0, but includes the βu. In the Legendre's associated equation, λ is constrained by eigenvalues of the form l(l+1), where l takes on positive integer values. I'll be looking around for solutions to the equation in question, but I would at least like confirmation as to whether λ here is constrained to integral values (in other words, the nature of the eigenspectrum). I suspect that they are not.