I've been asked to find the Riemann Invariants for the system: $$ \begin{pmatrix} \cos(v) & 0 \\ 0 & \cos(v) \end{pmatrix} \begin{pmatrix} u_x \\ v_x \end{pmatrix} + \begin{pmatrix} \sin(v) & -1 \\ -1 & \sin(v) \end{pmatrix} \begin{pmatrix} u_y \\ v_y \end{pmatrix} = 0$$
It's easy to show that the two families of characteristic projections satisfy:
$$\frac{dy}{dx} = \tan(v) \pm \sec(v)$$ and that along these curves we have, for $\underline{u} = (u,v)$:
$$(1,\mp1) (\cos(v)\frac{\partial \underline{u}}{\partial x} +(\sin(v)\pm1)\frac{\partial \underline{u}}{\partial y}) = 0 $$
I'm only really having trouble with the last step, which is finding the Riemann invariants from this (i.e. the $R_\pm$ such that the above expression can be written as $\frac{d}{dx}R = 0$.)
Any help in seeing what these Riemann invariants are would be appreciated.