  This article is concerned with the question: For which pairs of hyperbolic
Euler-Lagrange systems in the plane does there exist a rank-$1$ B\"acklund
transformation relating them? We express some obstructions to such existence in
terms of the local invariants of the Euler-Lagrange systems. In addition, we
discover a class of B\"acklund transformations relating two hyperbolic
Euler-Lagrange systems of distinct types.
