Integrable Generators of Lie Algebras of Vector Fields on textrm{SL}_2(mathbb {C}) and on xy = z^2

Abstract

For the special linear group textrm{SL}_2(mathbb {C}) and for the singular quadratic Danielewski surface x y = z^2 we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on x y = z^2.