Preliminary Classification of Realizations of Two-Dimensional Lie Algebras of Vector Fields on a Circle
Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical form of realizations of two-dimensional noncommutative algebra is obtained. It is shown that all other realizations of smooth vector fields are reduced to this form using global transformations.