The spectral problem for the generalized Zakharov-Shabat operator acting on N-vector functions on the line is studied. The eigenvalue problem is transformed to the Fredholm integral equations which yield the Jost solutions meromorphic on the half plane of the complex spectral parameter. Multiplying the Fredholm determinant on the J ost solution, we define the modified J ost solution which becomes regular on the half plane. Some formulas are obtained for the analysis of Fredholm determinants especially in connection with the analyticity problem. The resolvent operator and the expansion theorem for the differential operator are presented and the spectrum is obtained. The point spectrum is shown to be the subset of the poles of the Jost solutions, i.e., the set of zeros of the Wronskian of Modified Jost solutions.