Hecke cycles associated to rank 2 twisted Higgs bundles on a curve

Let $X$ be a smooth complex projective curve of genus $g$ and let $L$ be a line bundle on $X$ with $\mathrm{d}\mathrm{e}\mathrm{g}L>0$. Let $\mathbf{M}$ be the moduli space of semistable rank 2 $L$-twisted Higgs bundles with trivial determinant on $X$. Let ${\mathbf{M}}_X$ be the moduli space of stable rank 2 $L$-twisted Higgs bundles with determinant $\mathcal{O}(-x)$ for some $x\in X$ on $X$. We construct a cycle in the product of a stack of rational maps from nonsingular curves to ${\mathbf{M}}_X$ and ${\mathrm{P}\mathrm{i}\mathrm{c}}^{2\mathrm{d}\mathrm{e}\mathrm{g}L}(X)$ by using Hecke modifications of a stable $L$-twisted Higgs bundle in $\mathbf{M}$.