Summary: In this article, our task is to study the existence and Noethericity of solution for two classes of singular convolution integral equations with Cauchy kernels in the non-normal type case. To obtain the conditions of solvability for such equations, we establish regularity theory of solvability. By means of the theory of Fourier analysis, we will transform the equations into boundary value problems for holomorphic functions. The holomorphic solutions and conditions of solvability are obtained by using the method of complex analysis in class \(\{0\}\). Moreover, we also discuss the asymptotic property of solution near nodes. Therefore, our work generalizes and improves the theories of integral equations and the classical boundary value problems for holomorphic functions.