  We consider the prescribed scalar curvature problem on $ {\mathbb{S}}^N $ $$
\Delta_{{\mathbb S}^N} v-\frac{N(N-2)}{2} v+\tilde{K}(y) v^{\frac{N+2}{N-2}}=0
\quad \mbox{on} \ {\mathbb S}^N, \qquad v >0 \quad \mbox{on} \ {\mathbb S}^N,
$$ under the assumptions that the scalar curvature $\tilde K$ is rotationally
symmetric, and has a positive local maximum point between the poles. We prove
the existence of infinitely many non-radial positive solutions, whose energy
can be made arbitrarily large. These solutions are invariant under some
non-trivial sub-group of $O(3)$ obtained doubling the equatorial. We use the
finite dimensional Lyapunov-Schmidt reduction method.
