The asymptotics of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin metric on the singular locus: subintegrable systems

We study the asymptotic hyperk\"ahler geometry of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin moduli space over the singular fibers of the Hitchin fibration. We extend the previously known exponential convergence results for solutions to the Hitchin equation to the class of locally fiducial Higgs bundles defined by a special local description at the singularities of the spectral curve. This condition is satisfied by the Higgs bundles contained in certain subintegrable systems introduced by Hitchin. We prove that the restriction of the hyperk\"ahler metric to the subintegrable system converges exponentially fast to the corresponding semi-flat metric along a ray $(\mathcal{E},t\varphi)$. This answers a question posed by Hitchin in \cite{Hitchin2021subintegrable_special_Kaehler}. More generally, we prove that for each stratum of quadratic differentials there is a closed subset of the corresponding Hitchin fibers, such that the restricted hyperk\"ahler metric converges to a generalized semi-flat metric.