  The quantum $H_4$ integrable system is a 4D system with rational potential
related to the non-crystallographic root system $H_4$ with 600-cell symmetry.
It is shown that the gauge-rotated $H_4$ Hamiltonian as well as one of the
integrals, when written in terms of the invariants of the Coxeter group $H_4$,
is in algebraic form: it has polynomial coefficients in front of derivatives.
Any eigenfunctions is a polynomial multiplied by ground-state function
(factorization property). Spectra corresponds to one of the anisotropic
harmonic oscillator. The Hamiltonian has infinitely-many finite-dimensional
invariant subspaces in polynomials, they form the infinite flag with the
characteristic vector $\vec \al\ =\ (1,5,8,12)$.
