  Let $X,U,Y$ be spherically symmetric distributed having density $$\eta^{d
+k/2} \, f\left(\eta(\|x-\theta|^2+ \|u\|^2 + \|y-c\theta\|^2 ) \right)\,,$$
with unknown parameters $\theta \in \mathbb{R}^d$ and $\eta>0$, and with known
density $f$ and constant $c >0$. Based on observing $X=x,U=u$, we consider the
problem of obtaining a predictive density $\hat{q}(y;x,u)$ for $Y$ as measured
by the expected Kullback-Leibler loss. A benchmark procedure is the minimum
risk equivariant density $\hat{q}_{mre}$, which is Generalized Bayes with
respect to the prior $\pi(\theta, \eta) = \eta^{-1}$. For $d \geq 3$, we obtain
improvements on $\hat{q}_{mre}$, and further show that the dominance holds
simultaneously for all $f$ subject to finite moments and finite risk
conditions. We also obtain that the Bayes predictive density with respect to
the harmonic prior $\pi_h(\theta, \eta) =\eta^{-1} \|\theta\|^{2-d}$ dominates
$\hat{q}_{mre}$ simultaneously for all scale mixture of normals $f$.
