Rotations and Integrability

Abstract We discuss some families of integrable and superintegrable systems in $$n$$ -dimensional Euclidean space which are invariant under $$m\geqslant n-2$$ rotations. The invariant Hamiltonian $$H=\sum p_{i}^{2}+V(q)$$ is integrable with $$n-2$$ integrals of motion $$M_{\alpha}$$ and an additional integral of motion $$G$$ , which are first- and fourth-order polynomials in momenta, respectively.