A problem of reconstruction of the first kind discontinuities of a given in a unit disk B discontinuous function by its attenuated ray transform is considered. A mathematical model of an inhomogeneous medium includes known Riemannian metric and two positive functions. One of them describes a known absorption factor ϵ of the medium, and the other has the physical meaning of a distribution f of internal sources to be determined. The attenuated ray transform is the integrals of f with the weight depending on ϵ, which are evaluated along all geodesics in B, and are the initial data for the problem. The algorithm of approximate determination of lines of discontinuities of the function f having the discontinuities of the first kind is suggested and realized. The algorithm is investigated numerically. The results of simulation are satisfactory and allow to hope for the further development of the approach.