Rotations and integrability

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