Inverse spectral problem of reconstructing one-dimensional perturbation of integral Volterra operator

An integral operator representable as the sum of a Volterra operator and one-dimensional one is considered, when the inverse operator for Volterra one is an integro-differential operator of second order. The inverse problem of reconstruction of the one-dimensional item from spectral data provided that the Volterra component is known a priori is investigated. The uniqueness of the solution of the inverse problem is proved and conditions are obtained that are necessary and sufficient for its solvability.