  An effective family of spectral curves appearing in Hitchin fibrations is
determined. Using this family the moduli spaces of stable Higgs bundles on an
algebraic curve are embedded into the Sato Grassmannian. We show that the
Hitchin integrable system, the natural algebraically completely integrable
Hamiltonian system defined on the Higgs moduli space, coincides with the KP
equations. It is shown that the Serre duality on these moduli spaces
corresponds to the formal adjoint of pseudo-differential operators acting on
the Grassmannian. From this fact we then identify the Hitchin integrable system
on the moduli space of Sp(2m)-Higgs bundles in terms of a reduction of the KP
equations. We also show that the dual Abelian fibration (the SYZ mirror dual)
to the Sp(2m)-Higgs moduli space is constructed by taking the symplectic
quotient of a Lie algebra action on the moduli space of GL-Higgs bundles.
