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Algebraic-rational nature of the four-dimensional, $F_4$-invariant integrablequantum Hamiltonians, both rational and trigonometric, is revealed andreviewed. It was shown that being written in $F_4$ Weyl invariants, polynomialand exponential, respectively, both similarity-transformed Hamiltonians are inalgebraic form, they are quite similar the second order differential operatorswith polynomial coefficients; the flat metric in the Laplace-Beltrami operatorhas polynomial (in invariants) matrix elements. Their potentials are calculatedfor the first time: they are meromorphic (rational) functions withsingularities at the boundaries of the configuration space. Ground stateeigenfunctions are algebraic functions in a form of polynomials in somedegrees. Both Hamiltonians preserve the same infinite flag of polynomial spaceswith characteristic vector $(1, 2, 2, 3)$, it manifests exact solvability. Aparticular integral common for both models is derived. The first polynomialeigenfunctions are presented explicitly.

$F_4$ Quantum Integrable, rational and trigonometric models: space-of-orbits view