A method based on Hobson's theorem /1/ is proposed for constructing exact solutions of an integral equation of the first kind, defined on an elliptical area, with a power-law (polar) kernel and a polynomial right side. Solutions of this equation with different asymptotic expansions in the neighbourhood of the boundary ellipse are presented in an explicit closed form. The problem of the pressure of a stiff elliptical cylinder with arbitrary polynomial form for the base in an inhomogeneous elastic half-space ( v = const, E = E αx 3 α ) is considered as an illustration. Rostovtsev /2,3/ earlier indicated just the functional form of the unbounded solution of the equation mentioned, but did not obtain a relationship to define the constant coefficients in this solution in closed form. The solution for the case when the right side of the integral equation is a polynomial of zeroth power is given in /4/. Results of an investigation of integral equations of the first kind with power-law kernel defined on circular and strip areas are presented in /4,5/. Utilization of Hobson's theorem and the linear recurrence relations obtained below for the generalized potential factors of an elliptical disc, enables us, in addition to the rest, to get rid completely of the awkward apparatus introduced in /3/ for functions generated by Lamé ellipsoidal functions.