Consider a Randers space $(M,F)$ that is the solution of the zermelo's navigation problem associated to a wind $W$ which is homothety; $\mathcal{L}_Wh=\sigma h$, $\delta$ constant, on a Riemannian space $(M,h)$. Then the Randers geodesics can be found using Theorem 2 of Robles.

Now I am wondering if there is any way with which one can find the Randers geodesics of the Randers space ($\mathbb{R}^3,F$) which comes from putting the wind $W=(b+a\sin kx,0,0)$ on the Euclidean space $\mathbb{R}^3$. By using the equation of the geodesics it sounds quite difficult.

P.S.: here $a, b$ and $k$ are some constant.