  The quantum $H_3$ integrable system is a 3D system with rational potential
related to the non-crystallographic root system $H_3$. It is shown that the
gauge-rotated $H_3$ Hamiltonian as well as one of the integrals, when written
in terms of the invariants of the Coxeter group $H_3$, is in algebraic form: it
has polynomial coefficients in front of derivatives. The Hamiltonian has
infinitely-many finite-dimensional invariant subspaces in polynomials, they
form the infinite flag with the characteristic vector $\vec \al\ =\ (1,2,3)$.
One among possible integrals is found (of the second order) as well as its
algebraic form. A hidden algebra of the $H_3$ Hamiltonian is determined. It is
an infinite-dimensional, finitely-generated algebra of differential operators
possessing finite-dimensional representations characterized by a generalized
Gauss decomposition property. A quasi-exactly-solvable integrable
generalization of the model is obtained. A discrete integrable model on the
uniform lattice in a space of $H_3$-invariants "polynomially"-isospectral to
the quantum $H_3$ model is defined.
