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

Geometry of B\"acklund Transformations II: Monge-Amp\`ere Invariants