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It is shown that the $E_8$ trigonometric Olshanetsky-Perelomov Hamiltonian,when written in terms of the Fundamental Trigonometric Invariants (FTI), is inalgebraic form, i.e., has polynomial coefficients, and preserves two infiniteflags of polynomial spaces marked by the Weyl (co)-vector and $E_8$ highestroot (both in the basis of simple roots) as characteristic vectors. Theexplicit form of the Hamiltonian in new variables has been obtained both bydirect calculation and by means of the orbit function technique. It is shown atriangularity of the Hamiltonian in the bases of orbit functions and ofalgebraic monomials ordered through Weyl heights. Examples of firsteigenfunctions are presented.

Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariate Polynomials. III. $E_8$ case