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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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References:
 [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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