1. Vector logarithmic potential problems on equilibrium with different interaction between components (given by the interaction matrix) were first formulated and studied by Gonchar and Rahmanov (see 1-5) in connection with research on asymptotic analysis of Hermite-Pade approximations. Since then the area of applications of vector logarithmic potential problems has essentially broadened. One modern application is the distribution of the eigenvalues of random matrix ensembles, or more precisely, ensembles of multiple orthogonal polynomials (see 6). In these applications the vector logarithmic potential problem is considered in the presence of an external field. We briefly introduce the terminology that will be needed for stating our results. To simplify the notation we restrict ourselves to the case of two dimensions and the so-called Nikishin interaction matrix (see 7).