also excludes this “intuitive” guess used in alternative methods [@Ames1965].

Formally, applying the stretching transformation to means to check under which conditions the point transformations $$\bar \tau=\lambda^{a}\tau,~~~\bar r=\lambda^{b}r,~~~\bar \rho=\lambda^{c}\rho,~~~\bar u=\lambda^{d}u,\label{eq:stretch}$$ leave the system of PDEs invariant. Here, $a,b,c,d$ are constants and $\lambda \in \mathbb{R}$ is the group parameter. Invariance is found for $$a=b\nonumber,~c\neq0,~d=0.\nonumber$$ Hence, the solutions are arbitrary functions of the group invariants given by the similarity variable $$\xi\left(r,\tau\right)=\frac{r}{\tau},\label{eq:simvar}$$ the condition for the density $$\rho\left(r, \tau\right)=S\left(\xi\left(r, \tau\right)\right)r^{-\kappa}\label{eq:densdist}$$ with $\kappa\equiv b/c$ together with the condition for the velocity $$u\left(r, \tau\right)=u\left(\xi\left(r, \tau\right)\right). \label{eq:veltrans}$$ The above group invariant solutions () inserted into the PDEs (\[eq:origsys\_c,eq:origsys\_i\]) yields the basic system of ODEs[^5]

$$\begin{aligned}
   \frac{\mathrm{d}u}{\mathrm{d}\xi}&=\frac{u\alpha-\kappa\xi}{\xi\left(\left(\xi-u\right)^2-1\right)},\label{eq:odev}\\
   \frac{\mathrm{d}S}{\mathrm{d}\xi}&=\frac{Su\left(\alpha-\kappa\right)\left(\xi-u\right)-S\kappa}{\xi\left(\left(\xi-u\right)^2-1\right)}.\label{eq:odes}
  \end{aligned}$$

Alternatives to the group invariants emerge by a simple scaling with an arbitrary constant, but results in the same system \[eq:odev,eq:odes\] [@Ames1965]. Solutions to the PDE are mapped from solutions of the ODE with the invariants of the SG \[eq:simvar,eq:densdist,eq:veltrans\]. Moreover, invariance reduces the independent variable $\tau\in \mathbb{R}_0^{+}$ to a simple constant. The concerning auxiliary conditions are derived in \[initbnd\]. Realizing that the velocity equation \[eq:odev\] is decoupled from the function $S$ allows to investigate the behavior of the solution graphically (see \[sec:ssimsol\]).

Auxiliary conditions {#initbnd}
--------------------

The 