Pre research Questions

[Asadi]

(a):Let us denote by $g$ the Lie algebra of infinitesimal
transformation of Lie point symmetry of a nonlinear PDE system and $r
$ its radical which is solvable and thus according to the Lie-Bianchi
thereom, the PDE under consideration can be reduce to another PDE with
less independent variable. It is traditional to find optimal system of
Lie algebra $g$ and hence derive the all group invariant solutions.
Now there will be arised two question:
1. Is there any optimal system (since it is not unique!) consist of
elements in the radical subalgebra?
2.
Don't you think if we find the optimal system concerning the radical
subalgebra $r$ instead of $g,$ then one can better characterize group
invariant solution!



(b) What can you say about solvability or reductions of the group
invariant solution if you compute Killing form of an optimal system
concerning either full Lie algebra $g$ or the radical subalgebra $r.$


I must say that the statement of questions may be a bit not standard.
We will be happy to have your comments on that, either on the
statement or some example showing and illustrating the question and
answers.