Characteristic vector field ($a{\partial }_x+b{\partial }_y$)

For $a{u}_x+b{u}_y=c$, the projected vector field $(a,b)$ has the characteristic projections as its integral curves. The lifted vector field is $(a,b,c)$ in $(x,y,u)$ coordinates. A vanishing projected vector field can obstruct a classical solution even when the lifted vector field does not vanish.