the evolution equations invariant, which implies that the intensities satisfy the following relations, \[EqnTpiI\] Second, in general in the definitions , one can always take $\chi(x)\in[0,\pi/2]$. With that interval in mind, it is interesting to note that $\pi/2-\chi(x)\in[0,\pi/2]$. Hence, applying the transformation \[EqnTpi2\] leaves the evolution equations of the coupled $\chi(x)$ and $\Delta\phi(x)$ invariant while keeping the boundary condition on $\chi(x)$ in the appropriate interval. Therefore, it is easy to conclude that if $\chi(x)$ and $\Delta\phi(x)$ are solutions to , then from $\pi/2-\chi(x)$ and $\Delta\phi(x)\pm\pi$ are also solutions to . As before, this implies some conditions on the intensities which are here given by \[EqnTpi2I\] It is important to note that are true simply because the intensities do not care about the sum of the phases. Indeed, if $\Sigma\phi(x)$ had been important for the intensities, *i.e.* if it had not decoupled, then the relations would not be true because the transformation does not leave the differential equation for $\Sigma\phi(x)$ in invariant.

Mixed Initial States
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As discussed in the introduction, in the X-ray regime, production mechanisms in magnetars have very different origins for ALPs and photons. The former comes predominantly from nucleon-nucleon bremsstrahlung of ALPs [@Iwamoto:1984ir; @Iwamoto:1992jp; @Nakagawa:1987pga; @Nakagawa:1988rhp; @Raffelt:1996wa; 