  This paper extends widely the work in \cite{GT13}. Existence and
non-existence results of isoparametric functions on exotic spheres and
Eells-Kuiper projective planes are established. In particular, every homotopy
$n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with
2 points as the focal set, in strong contrast to the classification of
cohomogeneity one actions on homotopy spheres \cite{St96} ( only exotic
Kervaire spheres admit cohomogeneity one actions besides the standard spheres
). As an application, we improve a beautiful result of B\'{e}rard-Bergery
\cite{BB77} ( see also pp.234-235 of \cite{Be78} ).

Differential Geometry;Geometric Topology