  In this paper, I study the isoparametric hypersurfaces in a Randers sphere
$(S^n,F)$ of constant flag curvature, with the navigation datum $(h,W)$. I
prove that an isoparametric hypersurface $M$ for the standard round sphere
$(S^n,h)$ which is tangent to $W$ remains isoparametric for $(S^n,F)$ after the
navigation process. This observation provides a special class of isoparametric
hypersurfaces in $(S^n,F)$, which can be equivalently described as the regular
level sets of isoparametric functions $f$ satisfying $-f$ is transnormal. I
provide a classification for these special isoparametric hypersurfaces $M$,
together with their ambient metric $F$ on $S^n$, except the case that $M$ is of
the OT-FKM type with the multiplicities $(m_1,m_2)=(8,7)$. I also give a
complete classificatoin for all homogeneous hypersurfaces in $(S^n,F)$. They
all belong to these special isoparametric hypersurfaces. Because of the extra
$W$, the number of distinct principal curvature can only be 1,2 or 4, i.e.
there are less homogeneous hypersurfaces for $(S^n,F)$ than those for
$(S^n,h)$.
