The paper is concerned with a very specific inverse spectral problem. This problem is formulated as follows: Given the Weyl matrix , construct a differential operator associated with the higher-order differential equation
on the half-line. There are pointed out special fundamental systems of solutions of (1). Moreover, a uniqueness theorem for the inverse problem is presented. In this context, the main equation of the inverse problem is obtained. Conditions on the Weyl matrix and an algorithm of solution of the inverse problem complete the paper.