New superintegrable system on a sphere

We consider the motion of a material point on the surface of a sphere in the field of $2n+1$ identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [3], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional $N$-particle system discussed in the recent paper [13] and show that for the latter system an analogous superintegral can be constructed.