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A few physicists have recently constructed the generating compatibilityconditions (CC) of the Killing operator for the Minkowski (M) , Schwarzschild(S) and Kerr (K) metrics. They discovered second order CC, well known for M,but also third order CC for S and K. In a recent paper, we have studied thecases of M and S, without using specific technical tools such as Teukolskiscalars or Killing-Yano tensors. However, even if S($m$) and K($m,a$) aredepending on constant parameters in such a way that S $\rightarrow $ M when $m\rightarrow 0$ and K $\rightarrow$ S when $ a \rightarrow 0$, the CC of S donot provide the CC of M when $m \rightarrow 0$ while the CC of K do not providethe CC of S when $a\rightarrow 0$. In this paper, using tricky motivatingexamples of operators with constant or variable parameters, we explain why theCC are depending on the choice of the parameters. In particular, the onlypurely intrinsic objects that can be defined, namely the extension modules, maychange drastically. As the algebroid bracket is compatible with the {\itprolongation/projection} (PP) procedure, we provide for the first time all theCC for K in an intrinsic way, showing that they only depend on the underlyingKilling algebras and that the role played by the Spencer operator is crucial.We get K$<$S$<$M with $2 < 4 < 10$ for the Killing algebras and explain why theformal search of the CC for M, S or K are strikingly different, even thougheach Spencer sequence is isomorphic to the tensor product of the Poincar\'{e}sequence for the exterior derivative by the corresponding Lie algebra.

A Mathematical Comparison of the Schwarzschild and Kerr Metrics