This paper continues a series of papers on the problems of the construction theory of the generalized solutions of singular differential equations in Banach spaces and is focused on the investigation of Cauchy problem solvability in the class of distributions for a singular differential equation of the first order with variable operators coefficients. These problems in the case of the Fredholm operator by derivative can be reduced to the investigation of the system of integral Volterra equations of the first kind with the kernels of general types. Based on the results of solvability of these systems in the class of continuous functions and the theory of fundamental operator-functions of degenerated differential operators in Banach spaces, we can construct the generalized solutions of the considered problem in the class of generalized functions with left-bounded support. The connection between the obtained generalized solution and the continuous solution of this problem was investigated. The formula of the fundamental operator-function of the nonstationary differential operator of the first order in regular case was obtained.