$\alpha=2$ and $\kappa=1$. We find a saddle point (triangle) and an unstable improper node (dot) where the solution (solid line) connecting these points is a stable solution. The others are possible solutions, and the dash dotted are the singular lines.[]{data-label="fig:pp_a2k1"}](Figure06.pdf)

### General restrictions for  {#secgenrestrict}

We argue at the end of \[anasolIP\] that for $\kappa\geq1$ the RHCs yield no physically appropriate solutions for this regime. Moreover, we showed an easy method for solving without any restriction to $\kappa$. But there is at least one we need to mention. Regarding again, it is easy to spot and to prove that $u\left(\xi\right)=\xi$ (see end of \[anasolIP\]) is a singular solution. We can isolate the system parameters $\alpha, \kappa$ due to this solution and find $\kappa=\alpha+1$. This relation restricts the parameter $\kappa$ in dependence of the geometry $\alpha$ inasmuch as every singular solution violates the RHCs \[eq:RHCV\]. Hence, we are able to give the limit $$\kappa<\alpha+1.$$ for every possible solution. On a physical argumentation however, a finite initial mass is required to state a proper initial condition $\rho_0$ (). By solving the mass integral $\left(M=c\int_0^r\rho_0r'^{\alpha}\mathrm{d}r'\right)$ for and for the critical value $\kappa=\alpha+1-\epsilon$, $\epsilon\geq0$ we find $$M=cS_\infty\int_0^r\frac{\mathrm{d}r'}{r'^{1-\epsilon}}=\frac{cS_\infty}{\epsilon}r^{\epsilon}\label{eq:mass}$$ where $c$ is a constant depending on 