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 / Schrödinger 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 (1967 Nuovo Cimento A 10 1061–84) where only separable systems were found, and the need for its revision.