Characteristic determinant of a boundary value problem, which does not have the basis property

In this paper we consider a spectral 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 the characteristic determinant of the spectral problem with an integral perturbation of boundary conditions. We show that the set of kernels of the integral perturbation, under which absence of basis properties of the system of root functions persists, is dense in $L_2$.