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We study tuples of matrices with rigidity index two in $\Sp_4(\mathbb{C})$,which are potentially induced by differential operators of Calabi-Yau type. Theconstructions of those monodromy tuples via algebraic operations and middleconvolutions and the related constructions on the level differential operatorslead to previously known and new examples.

Some fourth order CY-type operators with non symplectically rigid monodromy