An affine interpretation of Bäcklund maps

We consider an affine interpretation of Bäcklund maps for second-order differential equations with an unknown function of two arguments. (Note that Bäcklund transformations represent a special case of Bäcklund maps.) Until now, no one has interpreted Bäcklund transformations as transformations of surfaces in a space different from the Euclidean one. In this paper we consider only the so-called Bäcklund maps of class 1. We represent solutions of differential equations as surfaces in an affine space with an induced connection defining a representation of zero curvature.
We prove that if a second-order differential equation admits a Bäcklund map of class 1, then for every solution of this equation there exists a congruence of straight lines in an affine space generated by tangents to the affine image of the solution. This congruence is an affine analog of the parabolic congruence in a Euclidean space. One can interpret a Bäcklund map as a transformation of surfaces in the affine space such that the affine image of the solution of the given differential equation is mapped to a certain boundary surface of the congruence.