  We complete previous results about the incompressible limit of both the
$n$-dimensional $(n\geq3)$ compressible Patlak-Keller-Segel (PKS) model and its
stationary state. As in previous works, in this limit, we derive the weak form
of a geometric free boundary problem of Hele-Shaw type, also called congested
flow. In particular, we are able to take into account the unsaturated zone, and
establish the complementarity relation which describes the limit pressure by a
degenerate elliptic equation. Not only our analysis uses a completely different
framework than previous approaches, but we also establish a novel uniform $L^3$
estimate of the pressure gradient, regularity \`a la Aronson-B\'enilan, and a
uniform $L^1$ estimate for the time derivative of the pressure. Furthermore,
for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that
the incompressible limit of the PKS model is unique. In addition, we establish
the corresponding incompressible limit of the stationary state for the PKS
model with a given mass, where, different from the case of PKS model, we obtain
the uniform bound of pressure and the uniformly bounded support of density.
