arbitrary accuracy. We chose the 4^th^ order Adams-Moulton scheme again (see \[secRHC\]) and start the integration at the discontinuity $\mathcal{U}_\mathrm{u,d}\left(\xi=\xi_0\right)$ up to the origin $\xi=0$ and to $\left.\xi\right|_\infty=1000$. Again we do not lose uniqueness because we use the BCs to compute the RHCs. In the following, all solutions are plotted in the physical variables $\rho\left(r,\tau\right),~u\left(r,\tau\right)$, whereas we solve the dimensionless equations (the speed of sound is set to $c_{\mathrm{s}}=1$).

To investigate the behavior of the solutions to subject certain values for $\kappa$ and BCs we emphasize some general properties by analyzing the phase diagrams. The nominator and denominator function () gives expressions for critical lines and critical points. We find a infinite or zero slope of the velocity field along the singular lines $$u\left(\xi\right)=\frac{\kappa\xi}{\alpha},~~u\left(\xi\right)=\xi\pm1,~~
        \xi=0,\label{eq:negsign}$$ where the negative sign in \[eq:negsign\] accounts for intersections in the positive half-plane as long as $\kappa\leq\alpha$. For $\kappa>\alpha$ the intersection is due to the function with the positive sign. The last condition in \[eq:negsign\] leads to diverging behavior of the solutions and therefore a nullcline. The positions of critical points are determined as the intersection of the singular lines at $$\mathcal{D}_1=\left(0,0\right),~~~\mathcal{D}_2=\left(\frac{\pm\alpha}{\alpha-\kappa},\frac{\pm\kappa}{\alpha-\kappa}\right).\label{eq:singpnt}$$ The second point depends on geometry and on the exponent of the initial 