Uniqueness of the solution to M.M. Lavrentiev’s equation with sources on a circle

A linear integral equation related to the coefficient inverse problem for a hyperbolic equation is considered. In the inverse problem, based on measurements of scalar wave fields scattered by an inhomogeneity, it is necessary to reconstruct the propagation velocity of the signal on the inhomogeneity. Probing fields are generated by point sources centered on a circle. We prove the unique solvability of the inverse problem with such an arrangement of sources under quite general assumptions on the choice of a variety of detectors. The relationship between the axial symmetry of the scattering data and the symmetry of the desired function relative to the same axis is established.