In this paper, the possibilities of constructing normal-regular solutions of degenerate systems obtained from Laurichella systems by limiting transition are studied. A number of important special cases of systems with solutions in the form of normally regular solutions are investigated. Some properties of such series are proved, the connections of these series with the newly introduced functions of V.I.Khudozhnikov are established. The necessary conditions for the existence of normally regular solutions of Horn-type systems consisting of two or more equations are also established. The constructed new solutions are generalizations of the well-known Horn and Humbert functions of two variables. The necessary and sufficient conditions for the existence of normally regular solutions are established using the concepts of rank and antirank. A modified Frobenius-Latysheva method is used to construct the solution.