The Absorption Theorem for the Beltrami-Vekua Normal Form

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar{z}}-\mu w_z+\mathcal{A}w+\mathcal{B}\bar{w}=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $\mathrm{\Theta }=\mathrm{\mid }\mathcal{B}{\mathrm{\mid }}^2\mathrm{/}(1-\mathrm{\mid }\mu {\mathrm{\mid }}^2)dxdy$ and its total mass are invariants under multiplicative gauges $w\mapsto \phi w$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=\phi v^{\mathrm{\prime }}+\psi {\bar{v}}^{\mathrm{\prime }}$, the complex gauges being the case $\psi \equiv 0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tilde{\phi }=-i\lambda \mathrm{/}(\phi -\psi )$, where $\lambda$ is the spectral root of the structure polynomial.