I'm interested in solutions to the generalised integral eigenfunction equation
$$ f(z^2,d-1)=2 \int_{z}^\infty \frac{f(r^2,d)}{\sqrt{1-\frac{z^2}{r^2}}}dr = \int_{z^2}^\infty \frac{f(y,d)}{\sqrt{y-z^2}}dy $$ for $z>0$, $d> 1$ which appears when marginalizing out a variable from a multi-variate probability distribution. Normally I would try differentiating to turn into a differential equation, but the weakly divergent integrand means I'm finding this difficult. The second form of integral above looks somewhat like a convolution, but the specific solutions I know (given below) aren't nice combinations of translationally invariant functions. The equations also look a little like Volterra equations, but a homogeneous version so I'm not sure how to apply an Adomian series solution.
From other analysis some specific examples of solutions are the Gaussian and t distributions, $$ f_\mathcal{N}(z^2,d)=\frac{{\rm e}^{-\frac{z^2}{2b^2}}}{\left(2 \pi b^2\right)^{d/2}}, \quad f_t(z^2,d)=\frac{\Gamma(\frac{d+\nu}{2})}{\Gamma(\nu/2)\left(\nu \pi b^2\right)^{d/2}}\left(1+\frac{1}{\nu}\frac{z^2}{b^2}\right)^{-(d+\nu)/2} $$ but I would like to find any other solutions that exist. I expect there is a good reference for this that I haven't found, so assistance in identifying this reference or a solution method would be very helpful.
Edit: this appears to be the Abel transform, and according to Bracewell (2000), p354 another solution is $$ f_B(z^2,d)=\frac{\pi^{-d/2}}{\Gamma(\frac{2+\nu-d}{2})}\left(a^2-z^2\right)^{(\nu-d)/2}\Theta(|a|-|z|) $$