Construction of an analog of the Fredholm theorem for a class of model first order integro-differential equations with a singular point in the kernel

In this work, integral representations of the manifold of solutions in terms of two arbitrary constants have been found for a model first order integro-differential equation with a singular point in the kernel. Although the kernel of this equation is not of the Fredholm type, the solution of this equation in the class of functions vanishing at $x=a$ has been found in an explicit form. It has been shown that the solution of the equation contains either two arbitrary constants or one arbitrary constant. Moreover, the case where the integro-differential equation has a unique solution has been revealed.
For the integro-differential equation, analogs of the Fredholm theorem have been constructed.
The existence of arbitrary constants in the general solution gives us chance to investigate some initial or boundary value problems for this equation. However, it is necessary to note that, in contrast to usual problems, these problems in our case are posed with different weights.
Correctness of the obtained results is verified with the help of detailed solutions of examples.
The method can be used for solving higher order model and non-model integro-differential equations with singular and supersingular kernels.