Solvable Lie algebra application I am starting to study Lie algebras and when I reached the notion of solvable Lie algebra, I tryed to find concrete applications ( in physics for exemple) and I couldn't find one.
For exemple, Solvable group are very important for the unsolvability of quintic equation ( and  by the way, it's the only application I know of them). 
In the same manner, can we find application for solvable lie algebras ?
 A: Besides Lie's theory of integration by quadrature for flows of vector fields, there is a completely different theory of solving rational coefficient linear differential equations by repeatedly integrating, taking logarithms, taking exponentials, and forming rational functions thereof: differential Galois theory. A nice introduction is: Camillo de Lellis, Il teorema di Liouville ovvero perche non esiste la primitiva di $e^{x^2}$. The upshot is that there is an invariant of any rational coefficient linear differential equation, its differential Galois group, and if the equation has general solution obtained by such repeated operations, then its differential Galois group is solvable.
A: There are actually lots of applications of solvable Lie algebras,  especially in the field of integrable systems where the solvabiltiy of Hamilton's equations of motion is frequently related to the solvability of the Lie algebra of the integrals of motion of the system. 
For example: 

for any Hamiltonian system on $\mathbb{R}^{2n}$ with its standard symplectic structure, if there are $n$ integrals of motion $F_i$, which are functionally independent and they form a solvable Lie algebra (under the Poisson bracket) -plus some technicalities on the structure constants- then the system can be integrated, on the level-set of the integrals, by quadratures.   

This is a classic result: see for example Perelomov's book, p. 34, 35 Theorem 2, the subsequent example and the references therein. Some authors call this result Lie's theorem (although i am not sure if this is a standard terminology). 
On the other hand, you can find lots of concrete examples in Integrable Hamiltonian systems on low-dimensional Lie algebras (here is the -free- Russian version). 
There are lots of other references as well. A descent coverage might require a fairly sized review article. 
A: I have read that one motivation of Lie when he worked on transformation groups was to build a similar theory for solvability of ODEs by quadrature to Galois theory for solvability of polynomials.
You can find some very concrete examples in this spirit in the paper: "Bryant, R, An introduction to Lie groups and symplectic geometry". I think Robert Bryant might have many interesting things to tell about this.
(I want to post this as a comment only, but I can't post comment yet)
