unstable improper node as a source with its first eigenvector $\boldsymbol{z}_1=\left(1,0\right)^{\mathrm{T}}$ giving the main direction for the solutions to swell. All other solutions leave this point in direction of $\boldsymbol{z}_2=\left(2,1\right)^{\mathrm{T}}$. The correct ones are hence given with the one connecting the singular points and else satisfying the outer BC \[eq:ubnd\]. The discontinuity must be located between the point $\mathcal{D}_2$ and the intersection between the solution and the singular line.

The same is true in the case of $\alpha=2$. However, we find an unstable degenerate improper node, meaning there exists just one eigenvalue with multiplicity two and just one eigenvector $\boldsymbol{z}_1=\left(1,0\right)^{\mathrm{T}}$. The solution starting at the point $\mathcal{D}_2$ and entering the origin is stable. The BC again yields the correct upstream solution.

All the solutions concerning different coordinates $\alpha\in\left\{0,1,2\right\}$ together with the constant initial density distribution are summarized in \[fig:solk0comb\]a-f. The upstream velocity remains zero up to the discontinuity caused by the nature of the singular points. Hence, the density is forced to behave similarly because the right hand side of vanishes for zero velocities.

![Phase diagram for spherical coordinates ($\alpha=2, \kappa=0$). The singular lines and points (a saddle (triangle) and an unstable improper node (dots)) correspond to those in the 