Geodesic
Finite-Dimensional Subalgebras of the Lie Algebra of Vector Fields on the Circle
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 338-342

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Finite-dimensional subalgebras of the Lie algebra $\mathrm {Vect}(S^1)$ of smooth tangent vector fields on the circle are considered that consist of analytic vector fields. It is proved that (up to an isomorphism) there are only three such subalgebras: a one-dimensional subalgebra, a two-dimensional noncommutative subalgebra, and a three-dimensional subalgebra isomorphic to $\mathrm {sl}_2(\mathbb R)$. 
@article{TM_2002_236_a34,
     author = {M. S. Strigunova},
     title = {Finite-Dimensional {Subalgebras} of the {Lie} {Algebra} of {Vector} {Fields} on the {Circle}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {338--342},
     publisher = {mathdoc},
     volume = {236},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2002_236_a34/}
}
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M. S. Strigunova. Finite-Dimensional Subalgebras of the Lie Algebra of Vector Fields on the Circle. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 338-342. http://geodesic.mathdoc.fr/item/TM_2002_236_a34/