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We prove a criterion for a general self adjoint differential operator of rank 7 to have its monodromy group inside the exceptional algebraic group G2(C). We then classify orthogonally rigid local systems of rank 7 on the punctured projective line whose monodromy is dense in the exceptional algebraic group G2(C). We obtain differential operators corresponding to these local systems under Riemann-Hilbert correspondence.

The classification of orthogonally rigid $G_2$-local systems and related differential operators