  Taking full advantage of two independent projectively equivalent metrics on
the ellipsoid leading to Liouville integrability of the geodesic flow via the
well-known Jacobi-Moser system, we disclose a novel integrable system on the
sphere $S^n$, namely the "dual Moser" system. The latter falls, along with the
Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally)
St\"ackel systems. Moreover, it is proved that quantum integrability of both
Neumann-Uhlenbeck and dual Moser systems is insured by means of the conformally
equivariant quantization procedure.
