On variants for solvability of a three-dimensional Volterra equation in quadratures

A linear integral equation of the general type with three independent variables, for which the special case of this equation is known, has been studied along with indication of certain variants of solving it in quadratures. In this paper, new variants of such cases have been found by developing the technique used previously by the author for similar equations with two independent variables.
In the first step of reasoning, a method has been suggested for reduction of the original integral equation to a differential one, the Goursat boundary values for which are calculated in quadratures under certain conditions. Then, two approaches have been applied to the Goursat problem: direct construction of its solutions based on the already known results; factorization of the left side of the differential equation by using the first- and second-order operators. Each of these approaches allows to single out a particular class of integral equations of the considered type that are solved in quadratures.