# Does solvability of Lie algebra have useful application in study of PDEs?

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

• I have thought of a similar question, although not undertaken any serious study of Lie algebra or its representations, so I am looking very much forward to an answer. – user305860 May 30 '16 at 11:38