We present a classical and quantum analysis of a particle confined in a paraboloidal cavity formed by two confocal paraboloids. Classically, the system is integrable and presents three independent constants of motion, namely, the energy, the z-component of the angular momentum, and a third dynamical constant associated with the paraboloidal geometry, which can be derived from the separability of the Hamilton-Jacobi equation. We derive analytical expressions for the actions, which allow us to determine the two conditions to obtain periodic trajectories. We classify these trajectories through the indices (s, t, ¿). The caustic paraboloids that bound the motion provide a complete geometric characterization of admissible trajectories. Quantum mechanically, separability of the Schrödinger equation in parabolic coordinates yields eigenmodes described by Whittaker functions. We determine the energy spectrum and identify degeneracies arising not only from azimuthal symmetry but also from specific cavity deformations.
