  In this work we explore the consequences of considering from the very
beginning the stationary and axisymmetric properties of the Kerr black hole as
one attempts to derive this solution through the Kerr-Schild ansatz. The first
consequence is kinematical and is based on a new stationary and axisymmetric
version of the Kerr theorem that yields to the precise shear-free and geodesic
null congruence of flat spacetime characterizing the Kerr solution. A
straightforward advantage of this strategy is that now the parameter $a$
appears naturally as associated to the conserved angular momentum of the
geodesics due to axisymmetry. The second consequence is dynamical and takes
into account the circularity theorem. In fact, a stationary-axisymmetric
Kerr-Schild ansatz is in general incompatible with the circularity property
warranted by vacuum Einstein equations unless the remaining angular dependence
in the Kerr-Schild profile appears fixed in a precise way. Thanks to these two
ingredients, the integration of the Einstein equations reduces to a simple
ordinary differential equation on the radial dependence, whose integration
constant is precisely the mass $m$. This derivation of the Kerr solution is
simple but rigorous, and it may be suitable for any textbook.
