Parameterizations of black-hole spacetimes beyond circularity

We discuss parameterizations of black-hole spacetimes in and beyond General Relativity view their symmetry constraints: within the class axisymmetric, stationary spacetimes, we propose a parameterization that includes non-circular both Boyer-Lindquist as well horizon-penetrating coordinates. show how existing parameterizations, which make additional assumptions (first, circularity; second, hidden constant motion), are included new parameterization. Further, explain why coordinates may be more suitable to parameterize deviations from Kerr geometry. Our investigation is motivated by our result regular, spinning proposed [1,2] non-circular. This particular deviation circularity can cusps, dent an asymmetry photon rings surrounding shadow. Finally, explore black holes, promote spin parameter function, find indications regularity cannot achieved this setting. strengthens case for regular holes based on promotion mass function.