for a constant initial density distribution ($\kappa=0$). Every possible solution leaving $\mathcal{D}_2=\left(1,0\right)$ either reaches its maximum $\xi$-value at the intersection with the singular line $u\left(\xi\right)=\xi-1$ and loses uniqueness afterwards, or never intersects at all. Further, the values from plotted as a function of the shock position $u_\mathrm{d}\left(\xi_0\right)$ (dotted line in \[fig:pp\_blast\_k0\]) lies always above the singular line, meaning that no solution exists referring to $\kappa=0$ satisfying the RHCs without losing uniqueness.

But there are further restrictions given. On the one hand, the strict upper limit \[eq:strict\_limit\] holds, meaning that the values are diverging for $\kappa\rightarrow3$. On the other hand, we derive a lower limit for $\kappa$ by searching for a removeable discontinuity (see \[fig:pp\_kink\]). This solution exists as also seen in \[fig:blast\_kdep\] meaning $\mathcal{U}_\mathrm{d}-\mathcal{U}_\mathrm{u}=0$ and hence showing just a kink. The linearization \[eq:spanalysis\] around $\mathcal{D}_2$ yields a starting point for the kinked solution. $\kappa$ is now varied numerically until the outer BC $u_{\infty}=0$ is satisfied. Hence, genuinely isothermal BW solutions in spherical coordinates only exist in the range $$0.60409\leq\kappa<3. \label{eq:restrkappa}$$

![Phase diagram for the EP ($\alpha=2, \kappa=0$). We spot a saddle point (triangle) and an unstable degenerate improper node (dots) where the solution (solid) connecting these points is stable. The 