Hmelek∗ ∗ Department of Theoretical Physics † Department of Experimental Physics Peoples’ Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, Russia, 117198 In this paper we study the static axisymmetric solutions of the vacuum Einstein equations. <...> The purpose of this paper is to obtain a static solution which turned out to be appropriate for describing the gravitational field around an axisymmetric mass distribution. <...> In this paper the method of singular sources os considered and some new applications are presented. <...> By mean of the method of singular sources it is possible to construct gravitational multipoles which generalize the Schwarzschild solution. <...> The obtained static vacuum axisymmetric generalization of the Schwarzschild solution near two points of horizon has coordinate singularities. <...> If one considers axially symmetric solutions of gravistatics, then construction of gravitational multipoles becomes ambiguous. <...> Key words and phrases: gravistatic, Schwarzschild solution, Einstein equation, Weyl metric, asymptotically flat. 1. <...> Introduction The vacuum static Einstein equations for the case of spherical symmetry were considered in 1916 by Schwarzschild [1] who obtained a solution which turned out to be appropriate for describing the gravitational field around a spherically symmetric mass distribution. <...> Further research in the field of exact solutions of the general relativity equations was appreciably influenced by the work of Weyl [1]. <...> Weyl’s static axially symmetric vacuum equations formed a basis for obtaining new exact solutions by Chazy and Curzon, Erez and Rozen, Gutsunaev and Manko [1]. <...> By means of the method of singular sources it is possible to construct gravitational multipoles which generalize the Schwarzchild solution. 2. <...> Basic Equations For axially symmetric vacuum static gravitational fields, the line element reduces to the Weyl metric [1] ds2 = 1 f [e2γ(dρ2 +dz2)+ρ2dϕ2]−fdt2, (1) where ρ, z, ϕ and t are Weyl’s canonical coordinates and time. <...> Method of Singular Sources The right-hand side of (4) contains zero though actually there should be a certain singular function characterizing the distribution of sources. <...> Dwight H. B. Tables of Integrals and Other Mathematical Data. — Fourth edition. — New York: The Macmillan Publishing Co., Inc., 1961. 1 <...>