  It has been well established that, in attraction-repulsion Keller-Segel
systems of the form\begin{equation*}
  \left\{
  \begin{aligned}
  u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla
w), \\
  \tau v_t &= \Delta v + \alpha u - \beta v,\\
  \tau w_t &= \Delta w + \gamma u - \delta w
  \end{aligned}
  \right. \end{equation*} in a smooth bounded domain $\Omega \subseteq
\mathbb{R}^n$, $n\in\mathbb{N}$, with Neumann boundary conditions and
parameters $\chi, \xi \geq 0$, $\alpha,\beta,\gamma,\delta > 0$ and $\tau \in
\{0,1\}$, finite-time blow-up can be ruled out in many scenarios given
sufficiently smooth initial data if the repulsive chemotaxis is sufficiently
stronger than its attractive counterpart. In this paper, we will go - in a
sense - a step further than this by studying the same system with initial data
that could already be understood as being in a blown-up state (e.g. a positive
Radon measure for the first solution component) and then ask the question
whether sufficiently strong repulsion has enough of a regularizing effect to
lead to the existence of a smooth solution, which is still connected to said
initial data in a sensible fashion. Regarding this, we in fact establish that
the construction of such a solution is possible in the two-dimensional
parabolic-parabolic system and the two- and three-dimensional
parabolic-elliptic system under appropriate assumptions on the interaction of
repulsion and attraction as well as the initial data.
