let $A$ be a circulant matrix over the integers, i was wondering if anyone knew of resources discussing the diophantine equation $\det(A) = 1$. So in the case of the circulant matrix being a $3 \times 3$ matrix, the diophantine equation would be $$x^3+y^3+z^3 -3xyz = 1$$ I have seen this specific equation solved, but i am asking about the general case) where $x,y,z$ is the first column/row of the matrix. I was able to find one [paper][1] but I am having trouble reading it and wanted to read other explanations. I would appreciate if i could be directed to other resources.
[1]:Chamberland, M. (2008). A natural extension of the Pythagorean equation to higher dimensions. The Ramanujan Journal, 16(2), 169–179. doi.org/10.1007/s11139-007-9107-8