Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras

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 $\bf E_6$ and $\bf E_7$. The involutions of the local integrals of motion are proved by means of the classical $R$-matrix approach.