Given the Diophantine equation $$ r^2-ds^2 = x^2-dy^2 = q, $$ (where $q$ is a potentially unknown integer, and certainly need not be $1$), the two solutions $(r,s)$ and $(x,y)$ are called equivalent if and only if there exist integers $t,u$ such that $t^2-du^2=1$ and $(x,y)=(rt+dsu,ru+st)$ [Robertson etc.].
What relationships and/or transformations exist between non-equivalent solutions? Has anyone done serious research on this, e.g. what does it mean if $\gcd(s,y)>1$?