A simple truncation approach for systematically constructing Lax pairs and Bäcklund transformations for integrable equations

This paper presents a systematic truncation method for constructing Lax pairs and B"acklund transformations of integrable nonlinear differential equations. By combining the mirror method and the WTC approach, we establish connections between Painlev\'e analysis, Riccati linearization, and Lax integrability. Our framework enables the derivation of symmetric auto-B"acklund transformations and Schlesinger transformations, offering a unified approach to understanding these systems. This approach yields a constructive algorithm that links geometric, algebraic, and analytical perspectives of integrability with significant implications for physical applications.