Devoted to the problem of regularity of solutions of quasilinear elliptic equations and systems. An exact class of elliptic systems has been identified for which the first boundary value problem has smooth solutions. The main results are obtained using a universal iterative process. Various applications in mechanics and electrical engineering are considered; provides numerical implementations of the specified process for specific tasks. A number of properties of solutions have also been established (Liouville type theorem, Korn inequality with weight, etc.).