Vali-e-Asr university of Rafsanjan Wavelet and Linear Algebra 2383-1936 12 2 2025 12 01 Mutual extraction of Bäcklund transformations and Lax representations for the Korteweg–de Vries equation 735248 10.22072/wala.2026.2082722.1487 EN Sayed Mohammad Hoseini Mathematics Dep. Vali-E-Asr University of Rafsanjan Abedini Mohajeri Reza ‎Vali-e-Asr Rafsanjan University Journal Article 2025 12 31 ‎In this paper‎, ‎we investigate the intrinsic relationship between Bäcklund transformations and Lax representations for the Korteweg–de Vries (KdV) equation‎. ‎Viewing the KdV equation within the framework of integrable hierarchies‎, ‎we analyze how its Bäcklund transformations encode the underlying spectral structure‎. ‎We demonstrate that the Lax pair of the KdV equation can be systematically derived from its Bäcklund transformations‎, ‎and conversely‎, ‎that the Bäcklund transformations can be reconstructed directly from the associated Lax representation‎. ‎This bidirectional correspondence clarifies the geometric and algebraic role of Bäcklund transformations as discrete symmetries of the KdV equation and highlights their interpretation as Darboux-type transformations acting on the spectral problem‎.