  A family of real Hamiltonian forms (RHF) for the special class of affine 1+1
- dimensional Toda field theories is constructed. Thus the method, proposed in
[Mikhailov;1981] for systems with finite number of degrees of freedom is
generalized to infinite-dimensional Hamiltonian systems. We show that each of
these RHF is related to a special Z_2-symmetry of the system of roots for the
relevant Kac-Moody algebra. A number of explicit nontrivial examples of RHF of
ATFT are presented.
