polar coordinate case \[fig:pp\_a1k0\] given by the dashed dotted lines. The thin solid curve denotes the separation of the domain by $\xi=0$. Thick lines starting at singular points are selected solutions. []{data-label="fig:pp_a2k0"}](Figure03.pdf)

![Phase diagram for polar coordinates with ($\alpha=1, \kappa=1$). There is just one saddle point (triangle) and the thick line receding the origin is the real solution, whilst the others are some possible ones. The singular lines again are dashed dotted, the vertical line departs the phase plane ($\xi=0$). []{data-label="fig:pp_a1k1"}](Figure04.pdf)

![Phase diagram for polar coordinates with ($\alpha=1, \kappa=1$). There is just one saddle point (triangle) and the thick line receding the origin is the real solution, whilst the others are some possible ones. The singular lines again are dashed dotted, the vertical line departs the phase plane ($\xi=0$). []{data-label="fig:pp_a1k1"}](Figure05.pdf)

### Polar coordinates and  {#sec_a1k1}

We show at the end of \[anasolIP\] that there are no possible solutions for $\alpha=0,~\kappa\geq1$, because the RHCs are never satisfied for that case. Hence, we start with the discussion for polar coordinates.

The singular lines are given by $u\left(\xi\right)=\left\{\xi \wedge \xi\pm1\right\}$ and plotted in \[fig:pp\_a1k1\]. However, there exists just one singular point $\mathcal{D}_1\left(0,0\right)$, a saddle point gathering the stable solution along the direction of 