let $a,b,c,d,x,y,k$ be all non-zero positive integers >1. If $a^2-b^2 \neq0$,how do you find all the pairs $(x,y)$ such that $k(a^2-b^2)=2(ax-by)$. I have found so far only solutions where $\gcd(x,y)>1$. Are there any other ones if $\gcd(x,y)=1$? if so, how do i find them?