  Algebraic-rational nature of the four-dimensional, $F_4$-invariant integrable
quantum Hamiltonians, both rational and trigonometric, is revealed and
reviewed. It was shown that being written in $F_4$ Weyl invariants, polynomial
and exponential, respectively, both similarity-transformed Hamiltonians are in
algebraic form, they are quite similar the second order differential operators
with polynomial coefficients; the flat metric in the Laplace-Beltrami operator
has polynomial (in invariants) matrix elements. Their potentials are calculated
for the first time: they are meromorphic (rational) functions with
singularities at the boundaries of the configuration space. Ground state
eigenfunctions are algebraic functions in a form of polynomials in some
degrees. Both Hamiltonians preserve the same infinite flag of polynomial spaces
with characteristic vector $(1, 2, 2, 3)$, it manifests exact solvability. A
particular integral common for both models is derived. The first polynomial
eigenfunctions are presented explicitly.
