The Hering plane and associated designs

We associate an affine plane ${\mathscr A}_f$ with any automorphism $f$ of the additive group of the field $F=\operatorname{GF}(q)$, where $q$ is odd, $F^*=\Delta\cup-\Delta$, and $\Delta=\bigl\{x^fx\mid x\in F^*\bigr\}$. We compute the ternar of the plane ${\mathscr A}_f$. A simple construction of the Hering plane in the case $q=27$, $x^f=x-\operatorname{Tr}x=-x^3-x^9$ and two designs associated with it are described in detail.