Let us say I find the characteristic lines of some easy PDE $a U_x + b U_y = 0$ to be $bx-ay=c$, where $b, a, c$ are constants. Now, we say the solution must be constant along those lines, so it HAS to be a function of $(bx-ay)$. We can write $U(x,y)=f(bx-ay)$ then.
Here's my question. If I have another function $u(x,y)$ that is a solution and is constant along those same lines, but does not have the same fixed value along each line as $f(bx-ay)$, what says that it HAS to be able to be written in the form $u(x,y)=f(bx-ay)$?