differential equations (ODEs) by linking the independent variables $\left(r, \tau\right)$ to one new independent variable $\xi\left(r, \tau\right)$. Second, the reduced set of ODEs must also yield a well-posed problem. This is achieved if the auxiliary conditions reduce to a well-posed boundary value problem, solely expressible in terms of the similarity variables [@Ames1965; @Bluman2002].

In general there are several ways to find transformation groups, reducing a PDE to an ODE for example with a dimensional analysis [@Sedov1946; @Sedov1959] or with the method of separation of variables [@Ames1965]. However, these methods may not provide self-similarity necessarily, and are based on some “intuitive” guess to determine the transformation group [@Ames1965; @Sedov1959].

In this work, we want to apply the general method of stretching transformations ST (or Birkhoff’s first method) [@Ames1965; @Logan2008]. The ST applied to (systems of) PDEs yields self-similar solutions if there exists a group of point transformations leaving the set of PDEs invariant. This is meant as a duality principle: the method only provides a transformation- or stretching group (SG) if similarity exists and the other way around. Hence, the striking features of the ST are its simplicity and clarity in determining invariance and the corresponding SG at once. This method 