Riemann invariants of a special hyperbolic system

Question

How can one compute the Riemann invariants of the following one dimensional hyperbolic system? $$\begin{pmatrix} u \\ v \end{pmatrix}_t + \begin{pmatrix} -v & -u \\ |v|-k & \mathrm{sgn}(v)u \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}_x = 0, \ $$ where $k \in \mathbb R$. I am stuck because the eigenvectors of the system $F =(f_1, f_2)$ and $G=(g_1,g_2)$ have both components which depend on both $u$ and $v$, so I cannot get the Riemann invariants $r_1,r_2$ from $\nabla r_1 = (-f_2,f_1)$ and $\nabla r_2 = (-g_2,g_1)$