Determining the kernel of the viscoelasticity equation in a medium with slightly horizontal homogeneity

Under study is the inverse problem of determining the two-dimensional kernel for a system of viscoelasticity equations in a medium with slightly horizontal homogeneity in a half-space. The direct initial-boundary value problem for the displacement function contains zero initial data and the Neumann condition of a special form. The field of displacements of medium points is given for $x_3 =0$ as additional information. We assume that the kernel decomposes into an asymptotic series, construct some method for determining the kernel with accuracy of $O(\varepsilon^2)$ where $\varepsilon$ is a small parameter, and prove the theorems of global unique solvability and stability of the solution to the inverse problem.