A first-order partial differential equation of the common type with initial data in Cartesian coordinates on an infinite length line

The Cauchy problem for a quasi-linear first order partial differential equation is studied in case when initial data is given on an infinite length smooth line with non-vertical gradient. A system in 15 integral equations, a solution of which gives a solution of the considered Cauchy problem in original coordinates, is constructed. Local solvability conditions, which do not include in itself assumptions about behavior of the characteristic lines, are presented in a theorem announced here.