the photon rays, so it must be performed with every quantity that is not constant all over the radiation field; in particular, this even applies to the constants of geodesic motion (which are only constant [*along the rays*]{}) unless they are same for [*all*]{} the rays. With our choice made in previous section, this means that one has to consider the energy $E$ to be a function of $(r,\theta)$, whereas ${\cal K}$ is left constant since it has been chosen to be the same for all photons. The conservation condition can thus be expressed as $$\label{Div(Phi2.p)=0,partial}
  (r^2\Phi^2 p^r)_{,r}\sin\theta+\epsilon^\theta\sqrt{\cal K}\,(\Phi^2\sin\theta)_{,\theta}=0$$ or after substitution for $p^r=({\sqrt{\cal K}}/{r})\,\sqrt{{r^2}/{b^2}-N^2}\,$ $$\left(r\,\Phi^2\,\sqrt{\frac{r^2}{b^2}-1+\frac{2M}{r}}\;\sin\theta\right)_{\!\!,r}+
  \epsilon^\theta(\Phi^2\sin\theta)_{,\theta}=0 \;.$$ Expanding the product derivative and dividing through, one finds $$\label{equation-for-Phi}
  \left(r\,\Phi^2\right)_{,r}
  +\frac{\epsilon^\theta\,(\Phi^2\sin\theta)_{,\theta}}{\sqrt{\frac{r^2}{b^2}-1+\frac{2M}{r}}\;\sin\theta}
  +\Phi^2\,\frac{\frac{r^2}{b^2}-\frac{r^3}{b^3}\,b_{,r}-\frac{M}{r}}
                {\frac{r^2}{b^2}-1+\frac{2M}{r}}
  =0 \,,$$ where $b=r_{\rm eq}(1-2M/r_{\rm eq})^{-1/2}$ according to the constraint (\[EK-constraint\]). Apparently it is correct to keep the $\epsilon^\theta$ sign in the equation in order to distinguish between the flux approaching the axis and its successor continuing after crossing the axis, since otherwise the $\theta$-derivative would jump across the axis due to the reversal of the $\partial x^\mu/\partial\theta$ orientation.

Interaction of a test particle with the radiation flux {#interaction}
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The aim of this paper is to check 