Electromagnetic waves in medium with strange polarization vector

Question

I have an exercise in Electromagnetic waves, basically to find the refractive index of a wave in a medium with polarization $\mathbf{P}=\alpha \nabla \times \mathbf{E}$. I used the Maxwell equations inside a material and assuming a plane wave, I got the following system of equations: \begin{align*} (-k^{2}+\frac{\omega^{2}}{c^{2}})E_{0y}=-i\alpha\mu_{0}k\omega^{2}E_{0x}\\ (-k^{2}+\frac{\omega^{2}}{c^{2}})E_{0x}=i\alpha\mu_{0}k\omega^{2}E_{0y}\\ \end{align*} where $E_{0x},E_{0y}$, are the x,y components of the magnitude of the plane EM wave. I solved the system by assuming circular polarization. This gave 2 possible indexes for $E_{0x}=\pm iE_{0y}$. My question is if this is the most generic solution I could get, I am not quite satisfied because I get two possible $n$ one for left and one for right handed polarizations. Although this is a question that came from an exercise, I would also like a general answer about the nature of such a material (if it exists). I find it very strange for a polarization to depend on the curl.