  A few physicists have recently constructed the generating compatibility
conditions (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 the
cases of M and S, without using specific technical tools such as Teukolski
scalars or Killing-Yano tensors. However, even if S($m$) and K($m,a$) are
depending 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 do
not provide the CC of M when $m \rightarrow 0$ while the CC of K do not provide
the CC of S when $a\rightarrow 0$. In this paper, using tricky motivating
examples of operators with constant or variable parameters, we explain why the
CC are depending on the choice of the parameters. In particular, the only
purely intrinsic objects that can be defined, namely the extension modules, may
change drastically. As the algebroid bracket is compatible with the {\it
prolongation/projection} (PP) procedure, we provide for the first time all the
CC for K in an intrinsic way, showing that they only depend on the underlying
Killing 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 the
formal search of the CC for M, S or K are strikingly different, even though
each Spencer sequence is isomorphic to the tensor product of the Poincar\'{e}
sequence for the exterior derivative by the corresponding Lie algebra.
