Algebraic integrability and minimality of Lie equations for transitive, finite dimensional, non-commutative pseudogroups

We provide an algebraic characterization of transitive, finite-dimensional algebraic Lie pseudogroups (or $\mathcal{D}$-groupoids) that are algebraic integrable, that is, isogenous to the action groupoid of an algebraic group action. Our approach is based on the differential Galois theory of rational connections. Under suitable hypotheses on the Lie algebra of the $\mathcal D$-groupoid, its algebraic integrability is equivalent to the triviality of the differential Galois group of its $\mathcal D$-Lie algebra. Furthermore, we investigate the structure of highly non-integrable $\mathcal{D}$-groupoids, demonstrating that if the differential Galois group of the linear differential equation of their $\mathcal D$-Lie algebra is large enough, then they are minimal in the sense that they admit no non-trivial sub-$\mathcal{D}$-groupoids of positive dimension.