GeometryInFields

Question: Let $(N,\Phi\oplus\Psi)$ be a Ferrero pair where $\phi^2\not=1$ for all $\phi\in\Phi\setminus\{1\}$ while $\psi^2=1$ for some $\psi\in\Psi\setminus\{1\}$. Can we do something similar to Clay's "Geometry in Fields"?

Question: What conditions can be given in this case so that the geometry of surfaces can be observed in this construction?

Problem: Find more examples for clays "Geometry in Fields" construction, and explore the properties of them.