  The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field
theories related to exceptional simple Lie algebras, and to the spectral theory
of the associated Lax operators. Real Hamiltonian forms are a special type of
"reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie
algebras. Examples of real Hamiltonian forms of affine Toda field theories
related to exceptional complex untwisted affine Kac-Moody algebras are studied.
Along with the associated Lax representations, we also formulate the relevant
Riemann-Hilbert problems and derive the minimal sets of scattering data that
determine uniquely the scattering matrices and the potentials of the Lax
operators.
