On Kerr black hole deformations admitting a Carter constant and an invariant criterion for the separability of the wave equation

In a previous work of ours, the most general family of Kerr deformations—admitting a Carter constant—has been presented. This time a simple, necessary and sufficient condition in order for the aforementioned family to have a separable Klein-Gordon equation is exhibited. In addition, we provide a solid theoretical foundation that the maximum number of free to be chosen radial functions is three.