  When ${\cal{D}}: E \rightarrow F$ is a linear differential operator of order
$q$ between the sections of vector bundles over a manifold $X$ of dimension
$n$, it is defined by a bundle map $\Phi: J_q(E) \rightarrow F=F_0$ that may
depend, explicitly or implicitly, on constant parameters $a, b, c, ...$. A
"direct problem " is to find the generating compatibility conditions (CC) in
the form of an operator ${\cal{D}}_1: F_0 \rightarrow F_1$. When ${\cal{D}}$ is
involutive, that is when the corresponding system $R_q=ker(\Phi)$ is
involutive, this procedure provides successive first order involutive operators
${\cal{D}}_1, ... , {\cal{D}}_n$ . Though ${\cal{D}}_1 \circ {\cal{D}}=0 $
implies $ad({\cal{D}}) \circ ad({\cal{D}}_1)=0$ by taking the respective
adjoint operators, then $ad({\cal{D}})$ may not generate the CC of
$ad({\cal{D}}_1)$ and measuring such "gaps" led to introduce extension modules
in differential homological algebra. They may also depend on the parameters.
When $R_q$ is not involutive, a standard {\it prolongation/projection} (PP)
procedure allows in general to find integers $r,s$ such that the image
$R^{(s)}_{q+r}$ of the projection at order $q+r$ of the prolongation
${\rho}_{r+s}(R_q) = J_{r+s}(R_q) \cap J_{q+r+s}(E)\subset J_{r+s}(J_q(E)) $ is
involutive but it may highly depend on the parameters. However, sometimes the
resulting system no longer depends on the parameters and the extension modules
do not depend on the parameters because it is known that they do not depend on
the differential sequence used for their definition. The purpose of this paper
is to study the above problems for the Kerr $(m, a)$, Schwarzschild $(m, 0)$
and Minkowski $(0, 0)$ parameters while computing the dimensions of the
inclusions $R^{(3)}_1\subset R^{(2)}_1 \subset R^{(1)}_1 =R_1 \subset
J_1(T(X))$ for the respective Killing operators.

General Physics