Abelian integral of Nuttall on the Riemann surface of the cubic root of the third degree polynomial

In the previous preprint "Geometry of Hermite–Pade approximants for system of functions $\{f,f^2\}$ with three branch points" a statement and general approaches to a problem on asymptotics of Hermite–Pade approximants for two analytic functions with three common branch points. This problem has an interest in connection with the Nuttall's conjecture, which states (in particularly) that an algebraic function of the third order appears as the Cauchy transform of the limiting measure of poles distributions of the approximants. At that preprint we carried out analysis of the appearing algebraic functions of genus zero. In this preprint we consider the main (from our point of view) case corresponding to the algebraic functions of genus one.