Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $ A^TA=\begin{pmatrix} a & b \\ b & \ddots \end{pmatrix}$, for $b>0$.

As a user observed in the solution of Part 1 of this question, $\mathcal{F}$ is closed under left and right multiplication by permutation matrices. Partition $\mathcal{F}$ into equivalence classes, by writing $A\sim B$ if $B$ is obtained from $A$ by a sequence of row and column permutations.

**Question 1:** Does every equivalence class of $\mathcal{F}$ contain a circulant matrix?

**Question 2:** If so, can the circulant matrix be chosen such that $a_{ij}\geq 0$ with equality iff $i=j$?

There is a natural geometric reformulation of this problem:

*Describe equidistant configurations of $n$ points on an $(n-1)$-sphere, subject to positivity constraints.*