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Using the Katz–Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured    P  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 pullbacks we construct families of    G  2   ‐connections which are not of these types.

Classification of rigid irregular G 2 ‐connections