In this paper we consider a spectral problem of one boundary value problem for a two-fold differentiation operator with an integral perturbation of boundary conditions of one type which are regular, but not strongly regular. The unperturbed problem has an asymptotically simple spectrum, and its system of eigenfunctions does not form a basis in L_2. We construct a characteristic determinant of the spectral problem with an integral perturbation of boundary conditions. We show that a set of kernels of the integral perturbation, under which absence of basis properties of the system of root functions persists, is dense in L_2.
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