Integrable generators of Lie algebras of vector fields on ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ and on $xy=z^2$

For the special linear group ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ and for the singular quadratic Danielewski surface $xy=z^2$ 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 $xy=z^2$.