The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, 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 [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 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 H3-invariants "polynomially"-isospectral to the quantum H3 model is defined.
Computing Polynomial Solutions and Annihilators of Integro-Differential Operators with Polynomial Coefficients
