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Remarks on a Lie theorem on the integrability of differential equations in closed form. (English. Russian original) Zbl 1095.34501
Differ. Equ. 41, No. 4, 588-590 (2005); translation from Differ. Uravn. 41, No. 4, 553-555 (2005).
The author proves the folloving theorem (main result): Let \(v_1(x), v_2(x),\dots ,v_n(x)\) be a set of vector fields linearly independent at all points in \(\mathbb{R}^n=\{x\}.\) It is supposed that these fields generate a solvable Lie algebra \(g\) with respect to the ordinary commutator \([\cdot,\cdot]:\, [v_k,v_j]=c_{1,j}^1v_1+ c_{2,j}^2v_2+\dots +c_{k,j}^kv_k,\) \(k,j=1,2,\dots, n.\) Then, each of the \(n\) differential equations \(\dot{x}=v_j(x),\) \(x\in \mathbb{R}^n,\) \(j=1,2,\dots, n,\) is integrable by quadratures.

MSC:
34A05 Explicit solutions, first integrals of ordinary differential equations
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[1] Chebotarev, N.G., Teoriya grupp Li (Theory of Lie Groups), Moscow, 1940.
[2] Olver, P.J., Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986. Translated under the title Prilozheniya grupp Li k differentsial’nym uravneniyam, Moscow: Mir, 1989.
[3] Kozlov, V.V., Simmetriya, topologiya i rezonansy v gamil’tonovoi mekhanike (Symmetry, Topology, and Resonances in Hamiltonian Mechanics), Izhevsk, 1995. · Zbl 0921.58029
[4] Kaplansky, I., An Introduction to Differential Algebra, Paris: Hermann, 1957. Translated under the title Vvedenie v differentsial’nuyu algebru, Moscow: Inostr. Lit., 1959. · Zbl 0089.02301
[5] Kozlov, V.V. and Kolesnikov, N.N., Vestn. Mosk. Univ. Mat., Mekh., 1979, no. 6, pp. 88–91.
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