Minimality of the $\mathcal{D}$-groupoid of symmetries of a projective structure

In this article we study Kummer’s $\mathcal{D}$-groupoid, which is the groupoid of symmetries of a meromorphic projective structure. We give necessary and sufficient conditions for its minimality, in the sense of not having infinite sub-$\mathcal{D}$-groupoids. The condition that we find turns out to be equivalent to the strong minimality of the non-linear Schwarzian equation and the non-integrability by means of Liouvillian functions of the linear Schwarzian equation.