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The GKZ system for the Hesse pencil of elliptic curves has more solutionsthan the period integrals. In this work we give different realizations andinterpretations of the extra solution, in terms of oscillating integral,Eichler integral, chain integral on the elliptic curve, limit of a period of acertain compact Calabi-Yau threefold geometry, etc. We also highlight the roleplayed by the orbifold singularity on the moduli space and its relation to theGKZ system.

GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities