Here we recall the well-known Chern–Terng theorem concerning affine minimal surfaces. After that we formulate some complementary (with transversal fields necessarily not parallel) affine Bäcklund theorem. Next, we describe some geometrical conditions which imply the local symmetry of both induced connections. Finally, we give some necessary and sufficient conditions under which the affine fundamental forms are proportional.
"On Analogues of Bäcklund Theorem in Affine Differential Geometry of Surfaces." J. Geom. Symmetry Phys. 54 79 - 110, 2019. https://doi.org/10.7546/jgsp-54-2019-79-110