Separation of variables for type $D_n$ Hitchin systems on hyperelliptic curves

Darboux coordinates for the Hitchin systems were known only in case of Lie algebra $\mathfrak{sl}(2)$ and a curve of genus 2, until recently. A description of the class of spectral curves for the Hitchin systems on hyperelliptic curves of all genera was given by O.K.Sheinman (2018). For Lie algebras of types $A_n$, $B_n$, $C_n$ the Darboux coordinates were found there in an explicit form. The main goal of my talk is to exhibit the difficulties arising in the case $D_n$, and find out Darboux coordinates for $n=2$ (Lie algebra $\mathfrak{so}(4)$) and genus two. The talk is a natural continuation of the talk by O.K. Sheinman, where the cases of Lie algebras $A_n$, $B_n$, $C_n$ are discussed.