The Cauchy Problem for the Systems ∂tz−∂x( y) = 0

In this paper we solve the Cauchy problem for the systems ∂tz−∂x γ = 0, where z = u + iv and γ ≤ 1 < 2. These systems are nonstrictly hyperbolic, possessing an isolated umbilic point at z = 0. We use the vanishing viscosity method with the help of the theory of compensated compactness. Uniform bounds in L∞ for the solutions of the viscous systems are not available, but such bounds can be found in L2. This makes necessary the use of the generalized Young measures and improvements in known techniques of the compensated compactness theory applied to conservation laws.