Symmetry-induced projective motions

Construct an incidence geometry with a symmetric realization that has a projective motion absent from realizations without that symmetry, so that the symmetric realization is more projectively flexible.

Background

The paper distinguishes ordinary flexibility from flexibility caused specifically by special geometric symmetry. It asks whether symmetry can create motions that disappear under nonsymmetric realizations of the same incidence geometry.

References

Does there exist an example of a configuration that has a symmetry-induced motion, that is, a motion which does not exist for a realization without that symmetry? In other words, can we find an example of an incidence geometry that has a symmetric realization, and a realization that does not have that symmetry, where the symmetric realization is more projectively flexible than the realization without the symmetry?

— Counting for rigidity under projective transformations in the plane  (2503.07228 - Berman et al., 10 Mar 2025) in Section "Open problems and future work," subsection "Symmetry-induced motions"