  We study a class of exceptional minimal surfaces in spheres for which all
Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy
\cite{ET0}, we obtain a description of exceptional surfaces in terms of a set
of absolute value type functions, the $a$-invariants, that determine the
geometry of the higher order curvature ellipses and satisfy certain Ricci-type
conditions. We show that the $a$-invariants determine these surfaces up to a
multiparameter family of isometric minimal deformations, where the number of
the parameters is precisely the number of non-vanishing Hopf differentials. We
give applications to superconformal surfaces and pseudoholomorphic curves in
the nearly K\"{a}hler sphere $S^{6}$. Moreover, we study superconformal
surfaces in odd dimensional spheres that are isometric to their polar and show
a relation to pseudoholomorphic curves in $S^{6}$
