Integrable Generators of Polynomial Vector Fields for Complex Classical Lie Groups

We consider in this paper complex Lie groups ${\mathrm{S}\mathrm{L}}_n(\mathbb{C})$ for $n\ge 2$, ${\mathrm{S}\mathrm{O}}_n(\mathbb{C})$ for $n\ge 5$, and ${\mathrm{S}\mathrm{p}}_{2n}(\mathbb{C})$ for $n\ge 1$. We show that for any such Lie group, the Lie algebra of polynomial vector fields can be generated by three $\mathbb{C}$-complete polynomial vector fields.