  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.
