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A new approach to the finite-gap property for the Heun equation isconstructed. The relationship between the finite-dimensional invariant spaceand the spectral curve is clarified. The monodromies are calculated and areexpressed as hyperelliptic integrals. Applications to the spectral problem forthe $BC_1$ Inozemtsev model are obtained.

The Heun Equation and the Calogero-Moser-Sutherland System III: The Finite-Gap Property and the Monodromy