We present the results from an observing campaign to confirm the peculiar motion of the supermassive black hole (SMBH) in J0437+2456 first reported in Pesce et al. (2018). Deep observations with the Arecibo Observatory have yielded a detection of neutral hydrogen (HI) emission, from which we measure a recession velocity of 4910 km s$^{-1}$ for the galaxy as a whole. We have also obtained near-infrared integral field spectroscopic observations of the galactic nucleus with the Gemini North telescope, yielding spatially resolved stellar and gas kinematics with a central velocity at the innermost radii ($0.1^{prime prime} approx 34$ pc) of 4860 km s$^{-1}$. Both measurements differ significantly from the $sim$4810 km s$^{-1}$ H$_2$O megamaser velocity of the SMBH, supporting the prior indications of a velocity offset between the SMBH and its host galaxy. However, the two measurements also differ significantly from one another, and the galaxy as a whole exhibits a complex velocity structure that implies the system has recently been dynamically disturbed. These results make it clear that the SMBH is not at rest with respect to the systemic velocity of the galaxy, though the specific nature of the mobile SMBH -- i.e., whether it traces an ongoing galaxy merger, a binary black hole system, or a gravitational wave recoil event -- remains unclear.
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