Solvability of the Hamiltonians related to exceptional root spaces: rational case

Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and $E_{6,7,8}$ is established in 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 {\it arbitrary} values of the coupling constants. The Hamiltonian of each model in algebraic form can be expressed as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators in a finite-dimensional representation