The study of the differential-algebraic boundary value problems was established in the
papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E.
Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi-
chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic
boundary value problems. At the same time, the study of differential-algebraic boundary-value
problems is closely related to the study of linear boundary-value problems for ordinary di-
fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N.
Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky,
I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk.
The study of the linear differential-algebraic boundary value problems is connected with
numerous applications of corresponding mathematical models in the theory of nonlinear osci-
llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the
actual problem is the transfer of the results obtained in the articles and monographs of S.
Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the
integro-differential boundary value problem not solved with respect to the derivative, in parti-
cular, finding the necessary and sufficient conditions of the existence of the desired solutions of
the linear integro-differential boundary value problem not solved with respect to the derivative.
In this article we found the conditions of the existence and constructive scheme for finding
the solutions of the linear Noetherian integro-differential boundary value problem not solved
with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian
integro-differential boundary value problem not solved with respect to the derivative in the
critical case in this article can be transferred to the seminonlinear integro-differential boundary
value problem not solved with respect to the derivative.