@Korobeinikov1976] first considered isothermality with the assumption of a vanishing temperature gradient ahead of the shock wave. This concept leads to an adiabatic flow through the discontinuity and the principle of energy conservation therefore holds. Together with boundary conditions given by adiabatic Rankine-Hugoniot Conditions (RHC) [@Rankine1870; @Hugoniot1887] well defined solutions exist. Nevertheless, above assumptions describe a mixed system of adiabaticity and isothermality and hence is not a genuinely isothermal system. Moreover, Lerche [@Lerche1977] stated that no valid isothermal self-similar blast-wave model provides applicability to SNR.

In this work, we want to give solutions of both the genuinely isothermal IP and EP with just one governing set of equations. In the investigated cases, the fluid will cool down immediately on both sides of the discontinuity to ensure constant temperature throughout the system. This does violate the principle of energy conservation but avoids the additional assumption of a vanishing temperature gradient ahead of the shock. We use similarity methods and give a mathematical justification for the existence and appearance of the Lie invariants via a stretching transformation. We derive one set of equations governing both, the IP and the EP by simply changing the corresponding boundary conditions. In case of the EP 