It is shown that the F 4 rational and trigonometric integrable systems are exactlysolvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)triangular. These variables are invariant with respect to the Weyl group of F 4 root system and can be obtained by averaging over an orbit of the Weyl group. Alternative way of finding these variables exploiting a property of duality of the F 4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasiexactly-solvable generalization of the rational F 4 model depending on two continuous and one discrete parameters is found.

Solvability of the rational quantum integrable systems related to exceptional root spaces G 2 , F 4 is re-examined and for E 6,7,8 is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitelygenerated Lie algebra of differential operators, taken in a finite-dimensional representation.

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