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

Abstract

AbstractFor the special linear group $$\textrm{SL}_2(\mathbb {C})$$ SL 2 ( C ) and for the singular quadratic Danielewski surface $$x y = z^2$$ 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 $$x y = z^2$$ x y = z 2 .