  It is shown that the $F_4$ rational and trigonometric integrable systems are
exactly-solvable for {\it arbitrary} values of the coupling constants. Their
spectra are found explicitly while eigenfunctions are obtained 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 a certain invariants of the $F_4$ Weyl group. Both Hamiltonians
preserve the same (minimal) flag of spaces of polynomials, which is found
explicitly.
