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We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is regular holonomic if G acts on X with finitely many orbits. When X is a partial flag variety, this construction gives the Picard-Fuchs system governing the period integrals on Calabi-Yau hypersurfaces in X. When X is a toric variety, our construction recovers certain generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky. Such systems have played important roles in the study of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.