The reference here is Pinchover & Rubinstein's An introduction to partial differential equations, pages 36-37. It's about the existence and uniqueness of a solution to the equation $a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)$ with initial condition (curve on the integral surface) $x(0,s)=x_0 (s), y(0,s)=y_0(s), u(0,s)=u_0(s)$.
It involves the construction of a surface, by solving a family of systems of ODEs (whose solutions $x(t,s), y(t,s), u(t,s)$) are curves on the desired surface and thus "knit it together"). They claim that "the transversality condition ($x_t(0,s)y_s(0,s)-y_t(0,s)x_s(0,s)\neq 0$) implies that the parametric representation provides a smooth surface". Why is it enough to check this along the initial curve?
Then they check that this surface satisfies the PDE. On they go: "to show that there are no further integral surfaces, we prove that the characteristic curves we constructed must lie on an integral surface". I don't know what this means, because I thought that was precisely what they had just done.
Most likely I'm just completely clueless and their writing is not to blame.
I appreciate any comments that may enlighten me on understanding this proof.