The Author considers the general canonic form for quasi linear hyperbolic systems with $r + 1$ independent variables $x, y_{1}, \ldots, y_{r}$, with m unknown functions $z_{1}, \ldots, z_{m}$, measurable coefficients and Lipschitzian data, and proves a new variant of the Cauchy problem for an infinite slab $0 \le x \le a$, of small width $a > 0$. The Author proves that it is possible to assign m suitable linear combinations of the m unknowns on distinct planes $x = a_{i}$, $0 \le a_{i} \le a$, provided a is sufficiently small.