On the incompleteness of the classification of quadratically integrable Hamiltonian systems in the three-dimensional Euclidean space

Abstract

We present an example of an integrable Hamiltonian system with scalar potential in the three-dimensional Euclidean space whose integrals of motion are quadratic polynomials in the momenta, yet its Hamilton-Jacobi / Schrodinger equation cannot separate in any orthogonal coordinate system. This demonstrates a loophole in the derivation of the list of quadratically integrable Hamiltonian systems in [Makarov et al., A systematic search for nonrelativistic systems with dynamical symmetries. Nuovo Cimento A Series 10, 52:1061-1084, 1967] where only separable systems were found, and the need for its revision.