Source-balanced Riemann invariant ($\nabla R\cdot S=0$)

For a smooth hyperbolic system $U_t+A(U)U_x=S(U)$, suppose $\nabla R$ is a left eigenvector of $A$ with eigenvalue $\lambda$. The chain rule gives $({\partial }_t+\lambda {\partial }_x)R=\nabla R\cdot S$. Hence the homogeneous invariant remains an exact Riemann invariant when its gradient annihilates the source. For $U=(u,v)$, $A=\left(\begin{array}{cc}v& u\\ u^{-3}& v\end{array}\right)$, $S=(-u^3,u)$ and $u>0$, the function $R=v-1/u$ has $\nabla R=(u^{-2},1)$, eigenvalue $v+1/u$, and $\nabla R\cdot S=0$. On its zero level, the other characteristic speed vanishes and $v_t=1/v$ along constant spatial positions. Source terms need not destroy every invariant.