  We consider an inverse spectral problem that consists in the recovery of the
differential expression coefficients for higher-order operators with separated
boundary conditions from the spectral data (eigenvalues and weight numbers).
This paper is focused on the most principal issue of the inverse spectral
theory, namely, on the necessary and sufficient conditions for the solvability
of the inverse problem. In the framework of the method of the spectral
mappings, we consider the linear main equation of the inverse problem and prove
the unique solvability of this equation in the self-adjoint case. The main
result is obtained for the first-order system of general form and so can be
applied to higher-order differential operators with regular and distribution
coefficients. From the theorem on the main equation solvability, we deduce the
necessary and sufficient conditions on the spectral data for a class of
arbitrary order differential operators with distribution coefficients. As a
corollary of our general results, we obtain the characterization of the
spectral data for the fourth-order differential equation in terms of
asymptotics and simple structural properties.
