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In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the {\it Vessiot structure equations}. In all these cases, they discovered that the {\it compatibility conditions} (CC) for the corresponding Killing operator were involving {\it a mixture of both second order and third order CC} and their idea has been to exhibit only a {\it minimal number of generating ones}. However, even if they exhibited a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. Using the formal theory of systems of partial differential equations and Lie pseudogroups, the purpose of this difficult computational paper is to provide new intrinsic differential and homological methods involving the Spencer operator in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools are now available as computer algebra packages., Comment: In this v2 we improve the search of generating compatibility conditions for the Killing operators used in general relativity and the corresponding differential sequences, by means of new differential homological methods. The existence of a partition 10=4+4+2 of the Ricci tensor questions the mathematical coherence of Einstein equations with group theory and formal integrability