Spectral curves and separation of variables for hyperelliptic Hitchin systems of types $A_n$, $B_n$, $C_n$

We give a description of families of spectral curves for Hitchin systems of types $A_n$, $B_n$, $C_n$ on hyperelliptic curves of all genera. On this basis, we develop the separation of variables scheme for those systems, and explicitly find out Darboux oordinates for them. The case $D_n$ will be considered in the related talk by P.Borisova.