  The construction of a family of real Hamiltonian forms (RHF) for the special
class of affine 1+1-dimensional Toda field theories (ATFT) is reported. Thus
the method, proposed in [1] for systems with finite number of degrees of
freedom is generalized to infinite-dimensional Hamiltonian systems. The
construction method is illustrated on the explicit nontrivial example of RHF of
ATFT related to the exceptional algebras E_6 and E_7. The involutions of the
local integrals of motion are proved by means of the classical R-matrix
approach.
