that it is completely determined once the solutions to the coupled $\chi(x)$ and $\Delta\phi(x)$ differential equations are known. Hence both $\Sigma\phi(x)$ and $\phi_\perp(x)$ are irrelevant in computing the intensities.

The initial states are determined at the surface of the magnetar from the boundary conditions at $x=1$. For the two-state system described by $\chi(x)$ and $\Delta\phi(x)$, pure initial states satisfy $\chi(1)=n\pi/2$ with $n\in\mathbb{Z}$, and the boundary condition for $\Delta\phi(1)$ must satisfy $\Delta\phi(1)=m\pi$ with $m\in\mathbb{Z}$ to avoid singularities. The two different choices of phase difference for pure initial states lead to the same intensities.

Indeed, it is straightforward to verify that the transformation leaves the evolution equations invariant. Therefore, for boundary conditions given by $\chi(1)=\chi_0$ and $\Delta\phi(1)=\Delta\phi_0$, the intensities verify Thus, for the pure ALP initial state $\chi_0=0$, the two different choices of phase difference, say $\Delta\phi_0=0$ and $\Delta\phi_0=\pi$, give the same intensities. For the pure parallel photon initial state $\chi_0=\pi/2$, the two different choices of phase difference, say $\Delta\phi_0=0$ and $\Delta\phi_0=\pi$ again, give the same intensities since are also invariant under the transformation $\chi(x)\to\chi(x)\pm\pi$. By using such arguments, more can be said about the intensities.

Indeed, two other transformations play an important role in the following. First, the transformation \[EqnTpi\] leaves 