Fuchsian equations and reduction of the Abelian differentials

In 2000-2003, K. Takemura and the author independently considered the Fuchsian equation with 4 singular points, and showed that that for some values of the coefficients, the solutions of this equation depend on the hyperelliptic function $\nu(\lambda)$. The function $\nu(\lambda)$ satisfies the equation of a hyperelliptic surface of genus $g$ with a single infinitely far point. The genus of the surface is simply calculated by the coefficients of the Fuchsian equation. The equation of a hyperelliptic surface can also be easily found by the coefficients of the Fuchsian equation.
Note that it is not difficult to transform this Heun equation into Schrodinger equation with finite-gap elliptic Darboux-Treibich-Verdier potential using variable substitution. The existence of a link with elliptic functions leads to the existence of reductions of the corresponding hyperelliptic integrals to elliptic ones.
Using the connection formulas between elliptic functions with coordinated lattices of periods, it is possible to construct elliptic finite-gap potentials with poles located not only in the half-periods of elliptic functions. By making inverse substitutions, one can obtain finite-gap Fuchsian equations with an increased number of singular points and new hyperelliptic integrals reducing to elliptic ones.
The second class of Fuchsian equations associated with Riemann surfaces arises when differentiating complete elliptic integrals by a parameter located at one of the branching points.Note that if there is a mapping of a Riemann surface to an elliptic one, then there are hyperelliptic differentials whose periods satisfy the same Fuchsian equation as the periods of elliptic differentials. This fact makes it possible to calculate hyperelliptic differentials that are reduced to elliptic ones.