We study positive entire solutions $u = u(x,y)$ of the critical equation
\begin{equation} \label{sceriffo} \Delta_x u+{(\alpha+1)^2}|x|^{2\alpha} \Delta_y u=-u^{({Q+2})/({Q-2})} \textrm{in}\mathbb{R}^{n}= \mathbb{R}^m\times\mathbb{R}^k, \end{equation} where $(x,y)\in \mathbb{R}^m\times \mathbb{R}^k$ , $\alpha>0$ , and $Q=m+k(\alpha+1)$ . In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a “spherical symmetry result for solutions”. In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution $u$ of (1), after a suitable scaling and a translation in the variable $y$ , the function $v(x)= u(x,0)$ satisfies the equation
\begin{equation} \label{pistolero} {\mathrm{div}\!_x}(p\nabla_xv )-q v=-p v^{({Q+2})/({Q-2})},|x|\lt 1,\end{equation}
with a mixed boundary condition. Here, $p$ and $q$ are appropriate radial functions. In the last part, we prove that if $m=k=1$ , the solution of (2) is unique and that for $m\ge 3$ and $k=1$ , problem (2) has a unique solution in the class of $x$ -radial functions