  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 forms of realizations of two- and
three-dimensional noncommutative algebras are obtained. It is shown that all
other realizations of smooth vector fields are reduced to this form using
global transformations. Some combinatorial formulas for the number of
inequivalent realizations of these algebras are obtained.
