  In this paper, we explore the structure of the Hitchin morphism for higher
dimensional varieties. We show that the Hitchin morphism factors through a
closed subscheme of the Hitchin base, which is in general a non-linear subspace
of lower dimension. We conjecture that the resulting morphism, which we call
the spectral data morphism, is surjective. In the course of the proof, we
establish connections between the Hitchin morphisms for higher dimensional
varieties, the invariant theory of the commuting schemes, and Weyl's
polarization theorem. We use the factorization of the Hitchin morphism to
construct the spectral and cameral covers. In the case of general linear groups
and algebraic surfaces, we show that spectral surfaces admit canonical finite
Cohen-Macaulayfications, which we call the Cohen-Macaulay spectral surfaces,
and we use them to obtain a description of the generic fibers of the Hitchin
morphism similar to the case of curves. Finally, we study the Hitchin morphism
for some class of algebraic surfaces.
