Single-Valued Killing Fields of a Meromorphic Affine Connection and Classification

We give a geometric condition on a meromorphic affine connection for its Killing vector fields to be single valued. More precisely, this condition relies on the pole of the connection and its geodesics, and defines a subcategory. To this end, we use the equivalence between these objects and meromorphic affine Cartan geometries. The proof of the previous result is then a consequence of a more general result linking the distinguished curves of meromorphic Cartan geometries, their poles and their infinitesimal automorphisms, which is the main purpose of the paper. This enables to extend the classification result from [Biswas I., Dumitrescu S., McKay B., Épijournal Géom. Algèbrique 3 (2019), 19, 10 pages, arXiv:1804.08949] to the subcategory of meromorphic affine connection described before.