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Parameterized Kerr spacetimes allow us to test the nature of black holes inmodel-independent ways. Such spacetimes contain several arbitrary functionsand, as a matter of practicality, one Taylor expands them about infinity andkeeps only to finite orders in the expansion. In this paper, we focus on theparameterized spacetime preserving Killing symmetries of a Kerr spacetime andshow that an unphysical divergence may appear in the metric if such atruncation is performed in the series expansion. To remedy this, we redefinethe arbitrary functions so that the divergence disappears, at least for severalknown black hole solutions that can be mapped to the parameterized Kerrspacetime. We propose two restricted classes of the refined parameterized Kerrspacetime that only contain one or two arbitrary functions and yet canreproduce exactly all the example black hole spacetimes considered in thispaper. The Petrov class of the parameterized Kerr spacetime is of type I whilethat for the restricted class with one arbitrary function remains type D. Wealso compute the ringdown frequencies and the shapes of black hole shadows forthe parameterized spacetime and show how they deviate from Kerr. The refinedblack hole metrics with Kerr symmetries presented here are practically moreuseful than those proposed in previous literature.

Regularizing Parameterized Black Hole Spacetimes with Kerr Symmetries