Characteristic projection

For the first-order partial differential equation $a(x,y,z)z_x+b(x,y,z)z_y=c(x,y,z)$, a lifted characteristic curve satisfies $(x^{\prime },y^{\prime },z^{\prime })=(a,b,c)$. Its characteristic projection is the image $(x(s),y(s))$ in the plane of independent variables. The projection need not be a regular curve: if $a=b=0$, it can be stationary while $z$ varies. In that situation the lifted curve cannot lie in the graph of a single-valued classical solution unless the required compatibility holds. Noncharacteristic Cauchy data require the tangent to the initial curve and the projected characteristic vector to be linearly independent.