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

  • $\begingroup$ 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. $\endgroup$ – user305860 May 30 '16 at 11:38

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