We analyze two-dimensional systems related to the Helmholtz equation that allow separation of variables in both polar and parabolic coordinates. We pay special attention to the symmetry algebras involved in the separation of variables. We show how the modification of symmetry operators can lead from purely geometric symmetries to other dynamical ones, that is, from free systems to interacting systems, with the addition of potentials, which in our case are of two types: Kepler–Coulomb and Makarov. We also calculate the spectrum and associated eigenfunctions of the corresponding quantum mechanical systems, and we present a discussion of naturally separable classical systems, including the analysis of different types of trajectories. A discussion of the global properties of polar and parabolic coordinates is included, the relevance of which is demonstrated in the spectral and classical properties of these systems.
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