isothermal systems may lead to applicability to early phase SNR. Whilst many modern CFD codes are also able to model isothermal systems, corresponding test cases are merely rare. The results derived in this paper may expand the field of applicable test cases by a set of new solutions.

We start the investigation by describing the model assumptions and the derivation of the Lie invariants together with required auxiliary conditions in \[sec:matcons\]. The solutions are then discussed in \[sec:ssimsol\] by investigating the phase diagrams with respect to different geometries and free parameters to the system. The results are summarized in \[sec:summary\] and the \[anasolIP,anasolEP\] give analytical solutions to the IP and EP in planar geometry.

![Simplest pictures of 1D shocks in different coordinates. Figure (a) shows a shock front in plane geometry for a shock arising by fluid reflection at the wall (IP). Figure (b) shows the same in polar and spherical coordinates with reflecting boundary conditions at the center. The down ($\mathrm{d}$)- and upstream ($\mathrm{u}$) regions are labeled with respect to the shock front and change to $\left(d, u\right)^\mathrm{IP}\mapsto\left(u, d\right)^\mathrm{EP}$ while considering explosion problems. In this case, the explosion propagates into a quiescent region with $u\left(r\rightarrow\infty, t\right)=0$. []{data-label="fig:Figtube"}](Figure01.pdf)

Mathematical considerations 