In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are given in terms of the matrizant of the system in question. Recent developments in the theory of continuous analogs of the resultant operator play an essential role.
D. Alpay, I. Gohberg, M.A. Kaashoek, L. Lerer and A. Sakhnovich. Krein systems. Operator Theory: Advances and Applications, Vol. 191 (2009), 19-36.