The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

We consider the generalized Cauchy problem with data on two surfaces for a second-order quasilinear analytic system. The distinction of the generalized Cauchy problem from the traditional statement of the Cauchy problem is that the initial conditions for different unknown functions are given on different surfaces: for each unknown function we pose its own initial condition on its own coordinate axis. Earlier, the generalized Cauchy problem was considered in the works of C. Riquier, N. M. Gyunter, S. L. Sobolev, N. A. Lednev, V. M. Teshukov, and S. P. Bautin. In this article we construct a solution to the generalized Cauchy problem in the case when the system of partial differential equations additionally contains the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes. The last circumstance is a principal distinction of the problem in the present article from the generalized Cauchy problems studied earlier.