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