SIMPLIFICATIONS OF LAX PAIRS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS BY GAUGE TRANSFORMATIONS AND (DOUBLY) MODIFIED INTEGRABLE EQUATIONS

The paper’s Theorem 1 states that if M(u0,…,uk,λ) satisfies ∂_{ui0}( (∂_{ujk}M) M^{-1})=0 for all i,j, then there exists a gauge g(u0,…,uk−1,λ) such that M̂=S(g) M g^{-1} is independent of uk; the authors prove this by fixing a constant a0, choosing g=S^{-1}(M(a0, u1,…,uk,λ)^{-1}), and showing via equations (21)–(24) that ∂_{ujk}M̂=0, hence eliminating uk (Theorem 1 and its proof steps are in the cited passages). This argument is correct and complete in the paper’s framework, using the standard gauge action (9) and a connected domain hypothesis to conclude L_j≡0 from its u0-independence and vanishing at u0=a0 (see Theorem 1 statement and proof around equations (17), (18)–(24) and the definition of gauge action (9) . The candidate solution proves the same claim by a different route: it forms A_j=(∂_{ujk}M)M^{-1}, observes the Maurer–Cartan flatness, solves ∂_{ujk}g_+=−g_+A_j with initial data independent of u0 to obtain g_+(u1,…,uk,λ), sets g=S^{-1}(g_+), and then verifies ∂_{ujk}(S(g) M g^{-1})=0. This is a standard and locally valid constructive proof under the same hypothesis, and matches the paper’s conclusion. Therefore, both are correct and consistent, though they use different constructions.