Geometric Perspective of Reciprocal Transformations Induced by Conservation Laws in Integrable Systems

This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations. By interpreting conservation laws as closed 1-forms, we formalize reciprocal transformations as induced local diffeomorphisms on the jet bundle. This allows us to characterize higher-dimensional deformations as systematic fiber bundle extensions, where fiber coordinates are generated by potential functions of the conservation laws. This perspective provides an interpretation for the covariant lifting of Lax pairs to higher dimensions and reveals that auto-B{"a}cklund transformations are composite diffeomorphisms. These results are applied to several classical integrable models.