Let $\mathcal{F}$ denote the family of real [normal][1] 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][2], $\mathcal{F}$ 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][3] matrix?

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

----------
[Part 1 of this question][5]

[Original question on math.SE][6]

[Literature][4]

There is a natural geometric reformulation of this problem:

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


  [1]: http://en.wikipedia.org/wiki/Normal_matrix
  [2]: https://mathoverflow.net/questions/193510/is-a-normal-matrix-satisfying-ata-circulant/193723
  [3]: http://en.wikipedia.org/wiki/Circulant_matrix
  [4]: http://www.math.ohiou.edu/~jain/073.pdf
  [5]: https://mathoverflow.net/questions/193510/is-a-normal-matrix-satisfying-ata-circulant/193723
  [6]: https://math.stackexchange.com/questions/1092185/is-a-normal-matrix-satisfying-ata-circulant