  In this paper, we revisit the Kepler problem with linear drag. With
dissipation, the energy and the angular momentum are both decreasing, but in
\cite{margheri2017a} it was shown that the eccentricity vector has a
well-defined limit in the case of linear drag. This limiting eccentricity
vector defines a conserved quantity, and in the present paper, we prove that
the corresponding invariant sets are smooth manifolds. These results rely on
normal form theory and a blowup transformation, which reveals that the
invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic
orbits. Moreover, we identify a separate invariant manifold which corresponds
to a zero limiting eccentricity vector. This manifold is obtained as a
generalized center manifold over the zero eigenspace of a zero-Hopf point.
Finally, we present a detailed blowup analysis, which provides a geometric
picture of the dynamics. %We will use this to shed light on a degenerate case
of the limiting eccentricity vector. %The aforementioned invariant manifolds
only make up a subset hereof. We believe that our approach and results will
have general interest in problems with blowup dynamics.
