On the reduction of the degree of differential equation. The quadratic Darboux case

It is well known that Abelian integrals satisfy the Linear ODE of Fuchs type. The order of quation (generically) coincide to the dimension of the first homology group of the respective complex curve. In some cases it is possible to construct the lower order equation with algebraic coefficients. I will speak about the quadratic Darboux special case of this general phenomenon.