Distribution of integral points on second-order and certain third-order surfaces

Under certain assumptions regarding the bounds of the zeros of the Dirichlet L -functions, one obtains results on the asymptotics of the number of integral points in arbitrary domains on second-order surfaces of an arbitrary form. The method is based on reduction to the case of the simplest hyperboloids. As an application, one has obtained results on the distributions of the integral points on surfaces of the form $$x^3 + y^3 = u^2 + v^2 .$$