Symmetric tops in combined electric fields: Conditional quasisolvability via the quantum Hamilton-Jacobi theory

We make use of the quantum Hamilton-Jacobi (QHJ) theory to investigate conditional quasisolvability of the quantum symmetric top subject to combined electric fields (symmetric top pendulum). We derive the conditions of quasisolvability of the time-independent Schrödinger equation as well as the corresponding finite sets of exact analytic solutions. We do so for this prototypical trigonometric system as well as for its anti-isospectral hyperbolic counterpart. An examination of the algebraic and numerical spectra of these two systems reveals mutually closely related patterns. The QHJ approach allows us to retrieve the closed-form solutions for the spherical and planar pendula and the Razavy system that had been obtained in our earlier work via supersymmetric quantum mechanics as well as to find a cornucopia of additional exact analytic solutions