Abstract
The quantum H4 integrable system is a 4D system with rational potential related to the non-crystallographic root system H4 with 600-cell symmetry. It is shown that the gauge-rotated H4 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H4, is in algebraic form: it has polynomial coefficients in front of the derivatives. Any eigenfunction is a polynomial multiplied by ground-state function (factorization property). Spectra correspond to one of the anisotropic harmonic oscillators. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector α = (1, 5, 8, 12).
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