  In this work we construct an analytically completely integrable Hamiltonian
system which is canonically associated to any family of Calabi-Yau threefolds.
The base of this system is a moduli space of gauged Calabi-Yaus in the family,
and the fibers are Deligne cohomology groups (or intermediate Jacobians) of the
threefolds. This system has several interesting properties: the multivalued
sections obtained as Abel-Jacobi images, or ``normal functions'', of a family
of curves on the generic variety of the family, are always Lagrangian; the
natural affine coordinates on the base, which are used in the mirror
correspondence, arise as action variables for the integrable system; and the
Yukawa cubic, expressing the infinitesimal variation of Hodge structure in the
family, is essentially equivalent to the symplectic structure on the total
space.
