A Symmetry-Compatible Auxiliary-Field Splitting of the Korteweg–de Vries Equation: Exact Linearisation via the Direct and Inverse Pathways

This paper introduces a symmetry-compatible exact linearisation of the Korteweg–de Vries (KdV) equation through an auxiliary-field splitting based on Q=νfx. The term ‘symmetry’ is used here in a structural sense; the linear operator is held fixed across solution families, rather than in the sense of Lie point symmetry groups. The contribution is structural rather than solution-generating: Instead of proposing a new explicit KdV wave, we show that the nonlinear equation can be decomposed into a linear Airy-type equation for Q together with an algebraic reconstruction formula for f. The construction preserves the compatibility identity of KdV and transfers the distinction amongst invariant solution families to a source function Sx,t, whereas the linear operator remains fixed. The analysis identifies the self-consistency condition that distinguishes genuine KdV solutions from merely formal reconstructions on the direct pathway, and proves that the inverse pathway is exact for every sufficiently smooth KdV solution. We further characterise, in the travelling-wave sector, an explicit family of source functions for which the self-consistency condition holds a priori, turning the direct pathway into a constructive selection mechanism: The one-soliton emerges as an output rather than an input. The framework is illustrated by a source-driven selection of the one-soliton and by the inverse representation of classical invariant profiles, including a soliton on a constant background, the two-soliton interaction and the cnoidal wave. The resulting formulation places these structures within a single auxiliary-field framework and clarifies its relation to Bäcklund, Cole–Hopf, Miura and Darboux-type transformations.