How to solve $uu_{x_1}+u_{x_2}=1$ with characteristic method? $$uu_{x_1}+u_{x_2}=1\text{ with initial condition }u(x_1,x_1)=\frac{1}{2}x_1$$
I have problem to use following characteristic method to solve it.
Let $$x(s) = (x_1(s), x_2(s))$$$$z(s) = u(x(s))$$$$p(s) = (u_{x_1}(x(s)), u_{x_2}(x(s))$$$$F(x,z,p) = zp_1 + p_2 - 1 = 0$$
Assume, $$\frac{dx}{ds} = F_p = (z,1) $$$$\frac{dz}{ds} = F_p \cdot p = 1$$$$\frac{dp}{ds} = -F_x - F_z p = 0 - p_1(p_1,p_2) = - (p_1^2,p_1p_2)$$
Now, we can have, $$z(s) = s + c_0$$$$\dot x_1 = s + c_0 \Rightarrow x_1 = \frac{1}{2}s^2 + sc_0 + c_1$$$$\dot x_2 = 1 \Rightarrow x_2 = s + c_2$$
I'm stucked here. I don't know what to do next.

$uu_{x_{1}} + u_{x_{2}} = 1, u(x_{1}, x_{1}) = \frac{1}{2} x_{1}$
Let
\begin{cases}
      x(s) = (x_1(s), x_2(s))\\
      z(s) = u(x(s))\\
      p(s) = (u_{x_1}(x(s)), u_{x_2}(s))
\end{cases}
Need to solve the equation:
\begin{cases}
    \frac{dx_1}{ds} = z(s) \\
    \frac{dx_2}{ds} = 1 \\
\end{cases}
Observe that,
\begin{cases}
     \frac{dz}{ds} = 1
\end{cases}
Therefore, we have
\begin{cases}
     \frac{dx_1}{ds} = z(s) \\
        \frac{dx_2}{ds} = 1 \Rightarrow x_2 = s + c_0 \\
        \frac{dz}{ds} = 1 \Rightarrow z = s + z(0)
\end{cases}
where we let $x_1(0) = x_2(0) = c_0. $ and $z(0) = \frac{1}{2} c_0$
\begin{cases}
    \frac{dx_1}{ds} = z(s) \\
    z(0) = \frac{1}{2} c_0 \\
    z = s + z(0) \\
    x_1(0) = c_0
\end{cases}
$\Rightarrow x_1 = \frac{1}{2} s^2 + \frac{1}{2}c_0s + c_0$
Combine above, we now have
\begin{cases}
    x_1 = \frac{1}{2} s^2 + \frac{1}{2}c_0s + c_0 \\
    x_2 = s + c_0
\end{cases}
$\Rightarrow$
\begin{cases}
    s = \frac{2x_1 - 2 x_2}{x_2 - 2} \\
    c_0 = \frac{x_2^2 - 2 x_1}{x_2 - 2}
\end{cases}
$\text{Therefore, for point } (x_1,x_2) \in \mathbb{R}^2 \text{are at the line } x_{c_0}(s) \text{ coefficient } c_0 = \frac{x_2^2 - 2 x_1}{x_2 - 2} \text{ and parameter is } s = \frac{2x_1 - 2 x_2}{x_2 - 2} \text{, that is, } x_{c_0}(s) = (x_1,x_2).$
$\text{Also, we know that } u(x_1, x_2)=z(x_{c_0}(s) = s + \frac{1}{2}c_0 \text{. Then, } u(x_1,x_2) = \frac{x_1 - 2 x_2 + \frac{1}{2} x_2^2}{x_2 - 2} $
