  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 {\it 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 finitely-generated Lie algebra of differential operators, taken in a
finite-dimensional representation.
