If certain Lie algebra is solvable then what difference this algebra would create in application point view for PDEs ?
For example, in case of ODE of fourth order admitting three dimensional solvable Lie algebra $\mathfrak{g}$ can be reduce to quadrature three time in succession under ideals $L_{3}$, $L_{2}$, $L_{1}$, here
$L_{1}\subset L_{2}\subset L_{3}\subset \mathfrak{g}$
To be more precise, the ideal $L_{3}$ would reduce order of ODE from $4$ to $3$, the ideal $L_{2}$ would reduce order of ODE from $3$ to $2$ and ideal $L_{1}$ would further reduce order of ODE from $2$ to $1$ that is, reduction to quadrature.
Now this is really a very important aspect of solvable Lie algebra when we talk about ODEs, but what good solvable algebra can do for PDEs ?
Please refer to book "Symmetry and integration methods for differential equation" pp-$82$ for importance of solvability for ODEs
