A Schrödinger operator in \(m\) variables is said to be integrable if it has \(m\) algebraically independent quantum integrals which are commuting with each other. One of the most interesting integrable Schrödinger operators is the Calogero-Sutherland operator. An integrable Schrödinger operator is said to be algebraically integrable if the algebra of its quantum integrals cannot be generated by \(m\) operators. Calogero-Sutherland operators become algebraically integrable when the parameter takes special values. The authors study the algebraic integrability of certain matrix Schrödinger operators. To every complex simple Lie algebra \({\mathfrak g}\) and certain \({\mathfrak g}\)-module \(U\), a matrix Schrödinger operator is associated which is proved to be integrable [P. Etingof, J. Math. Phys. 36, No. 6, 2636-2651 (1995; reviewed above)].
The main result of this paper is that it is also algebraically integrable in the rational and trigonometric case if the \({\mathfrak g}\)-module \(U\) has highest weight. It is naturally conjectured to be true as well for the case of elliptic potential.