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.

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