Nuttall's decomposition of a three-sheeted Riemann surface of genus one

We investigate the structure of a decomposition of the Riemann surface $\mathfrak{R}$ of the function $\sqrt[3]{(z-a)(z-b)(z-c)}$ into $3$ sheets. The decomposition is specified by an Abelian integral with logarithmic singularities over the infinite points of $\mathfrak{R}$. In the case, when the triangle with vertices $a$, $b$, and $c$ is close to a regular one, the problem was studied by A. I. Aptekarev and D. N. Tulyakov. We consider the general case. The main attention is paid to investigation of the problem, if the critical points of the Abelian differential lie on the borders of the sheets.
The work is financially supported by the RFBR, grant No 18-41-160003.