Extremal measure and external field for two parameters vector equilibrium logarithmic potential problem

The vector equilibrium problem of the logarithmic potential theory in external eld with two independent parameters is considered. V. Buyarov and E. Rakhmanov have got explicit formulas that allow to nd the equilibrium measure, and to express the external eld through the family of equilibrium measures supports. These formulas are useful to obtain solutions of the continuum limit Toda lattice by means of the inverse problem method. Our goal is to obtain similar integral formulas for the vector case, where the vector measures are parameterized by the component masses.