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We present a detailed analysis of the behavior of the triaxial Schwarzschildorbit superposition method near the axisymmetric limit. Orbit superpositionmodeling is the primary method used to determine dynamical masses ofsupermassive black holes ($M_\mathrm{BH}$) in nearby galaxies; however, priorstudies have reported conflicting results when comparing the outcome fromaxisymmetric orbit codes with that from a triaxial orbit code in theaxisymmetric limit. We show that in order to achieve (oblate) axisymmetry in atriaxial code, care needs to be taken to axisymmetrize the short-axis tubeorbits and to exclude both the long-axis tube and box orbits from the orbitlibrary. Using up to 12 Gauss-Hermite moments of the line-of-sight velocitydistributions as constraints, we demonstrate the effects of orbit types on thebest-fit $M_\mathrm{BH}$ in orbit modeling of the massive elliptical galaxy NGC1453 reported in Liepold et al. 2020. In addition, we verify the efficacy ofour updated code on a mock galaxy dataset. We identify a subset of slowlyprecessing quasi-planar orbits for which the typical integration times can beinsufficient to fully capture the equilibrium orbital behavior in bothaxisymmetric and triaxial systems with central black holes. Furtherinvestigation is needed for a more reliable treatment of these orbits.

Dynamical Modeling of Galaxies and Supermassive Black Holes: Axisymmetry in Triaxial Schwarzschild Orbit Superposition Models