$\boldsymbol{x}_2=\left(2,1\right)^\mathrm{T}$. All three singular lines are parallel to each other. This is expressed in the diverging behavior of the singular point $\mathcal{D}_2$ \[eq:singpnt\] which therefore does not exist. The correct solution describing the IPs upstream part leaves the stable solution branch at some point, so that the BCs are satisfied. The jump location is just restricted by the area behind the intersection of the upstream solution with the singular line. The solutions are plotted in \[fig:figure7\]a,b. The velocity behind the origin rises rapidly up to the RHC and is caused by the geometry.

### Spherical coordinates and  {#a2k1}

In this case, the singular lines in \[fig:pp\_a2k1\] are given by $u\left(\xi\right)=\left\{\xi/2 \wedge \xi\pm1\right\}$. Solutions receding from the saddle point $\mathcal{D}_1=\left(0,0\right)$ are stable along the vector $\boldsymbol{x}_2=\left(3,1\right)^{\mathrm{T}}$ describing the downstream part. This solution originates from the unstable improper node $\mathcal{D}_2=\left(2,1\right)$ leaving in the main direction of $\boldsymbol{z}_2=\left(-1+\sqrt{5},1\right)^{\mathrm{T}}$. The second direction is given by $\boldsymbol{z}_1=\left(-1-\sqrt{5},1\right)^{\mathrm{T}}$. The upstream part again must reach the BC defining a intersection with the singular line. Again, the location of the discontinuity is restricted between this intersection and the improper node. The plots of the solutions are shown in \[fig:figure7\]c,d for different times $\tau$.

![Phase diagram for spherical coordinates 