The nonlocal solvability conditions for a system of quasilinear equations of the first order special right-hand sides

The Cauchy problem for a system of first-order quasilinear equations with special right-hand sides is considered. The study of solvability of this system in the original coordinates is based on the method of additional argument. It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions. For system of two equations non-local solutions are considered that are continued by finite number of steps from the local solution. Sufficient conditions for the existence of such non-local solution are derived. The proof of the non-local resolvability of the system relies on original global estimates.