Previously, for the two dimensional real moment problem, direct and inverse spectral problems were solved just as for the classical Hamburger moment problem. The result of the inverse spectral problem is block Jacobi type matrices. Direct problem consists of the recoveryng measures for a given of Jacobi type matrices. On the other hand, there were integrated Toda chains by spectral theory for the classical Hamburger moment problem using ordinary Jacobi matrix. In this article, using block Jacobi type matrices two-dimensional real moment problem, the system of Lax equations is constructed and integrated due to developed spectral theory for such matrices. For this reason the Weil function was presented there where recorded function that uniquely determined an appropriate measure correspondind to the two-dimensional real moment problem. For such functions there were pruved previously certain properties. Similar results for the complex moment problem were given by Berezansky Y.M. and Mokhonko O.A.