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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 localdoi:10.15507/2079-6900.20.201804.384-394 fatcat:oxnaak4d3zbflnu55ubosd6fmi