Classification of Rigid Irregular $G_2$-Connections

Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured ${\mathbb{P}}_{\mathbb{C}}^1$ having differential Galois group $G_2$, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pull-backs we construct families of $G_2$-connections which are not of these types.