Classical and generalized solutions of a mixed problem for a system of first-order equations with a continuous potential

A mixed problem for a first-order differential system with two independent variables and a continuous potential, the corresponding spectral problem for which is the Dirac system, is studied. Using a special transformation of the formal solution and refined asymptotics of the eigenfunctions, the classical solution of the problem is obtained. No excessive conditions on the smoothness of the initial data are imposed. In the case of an arbitrary square summable function, a generalized solution is obtained.