SO(4)-symmetry of mechanical systems with 3 degrees of freedom

We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra (Figure presented.) (4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie algebra (Figure presented.) (4), is presented. It is shown also that not every mechanical system with 3 degrees of freedom has first integrals generating (Figure presented.) (4).