A cases of solvability of the integral equation in quadratures

We consider Volterra equation with two-variable, commonly encountered in the theory of elasticity. The purpose is to find new variants of sufficient conditions for it's solvability in explicit calculation. The reduction principle of the original equation, first, to Goursat problem for differential equation of third order, and after that to two problems solving consecutively for equations of the first and second order is devised. One of these problems can be solved by direct equation integration, and the other's solution can be written through Riemann function for which variants of its explicit construction are found. Seven variants of conditions for mentioned calculation were obtained in terms of coefficients of the original equation. Considering that there are four variants of factorization of equation of third order involved into the reasoning, virtually there are 28 variants of conditions for original equation solvability in quadratures noted in this article.