The Coupled Hitchin-He Equations: Integrable Deformations and Rigidity of the Moduli Space

We introduce the \emph{parameter-geometrization} to the Hitchin system, a paradigm embedding deformation parameters into geometry via the coupled Hitchin-He equations on a surface with boundary. A boundary term couples a second Higgs field $\psi$, recovering the classical system at $\alpha=0$. We prove a unique, smooth solution branch exists near $\alpha=0$ (Theorem A). The system is integrable, admitting a Lax pair (Theorem B). Crucially, the moduli space $\mathcal{M}_\alpha$ is analytically isomorphic to $\mathcal{M}_0$ for small $|\alpha|$, preserving the Hitchin fibration -- revealing a deep rigidity where all moduli are controlled by the primary Higgs field (Theorem C). Using the \emph{nonlinear embedding} technique that casts the deformed system into the form of a classical Higgs bundle system, whose integrability and geometry are well-understood, we extends the framework to compact K\"ahler manifolds (Theorem D).