Rotations and Integrability

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.