  Every linear system of partial differential equations (PDEs) admits a scaling
symmetry in its dependent variables. In conjunction with other admitted
symmetries of linear type, the associated invariant solution condition poses a
linear eigenvalue problem. If this problem is structured such that the spectral
theorem applies, then the general solution of the considered linear PDE system
is obtained by summing or integrating the invariant eigenfunctions (modes) over
all eigenvalues, depending on whether the spectrum of the operator is discrete
or continuous. By first studying the 1-D diffusion equation as a demonstrating
example, this method is then applied to a relevant 2-D problem from
hydrodynamic stability analysis. The aim of this study is to draw attention to
the following two independent facts that need to be addressed in future studies
when constructing solutions for linear PDEs with the method of Lie-symmetries:
(i) Although each new symmetry leads to a mathematically different spectral
decomposition, they may all be physically redundant to standard ones and do not
reveal a new physical mechanism behind the overall considered dynamical
process, as incorrectly asserted, for example, in the recent studies by the
group of Oberlack et al. Hence, with regard to linear stability analysis, no
physically "new" or more "general" modes are generated by this method than the
ones already established. (ii) Next to the eigenvalue parameters, each single
mode can also acquire non-system parameters, depending on the choice of its
underlying symmetry. These symmetry-induced parameters, however, are all
physically irrelevant, since their effect on a single mode will cancel when
considering all modes collectively. In particular, the collective action of all
single modes is identical for all symmetry-based decompositions and thus
indistinguishable when considering the full physical fields.
