Given $a,b>0$ such that $a\ne b$ but nothing else is given (e.g. $a,b$ are not known to be coprime), I want to understand the structure of the set of solutions to the following system of two linear equations:
$a(x+y)+b(z+w)=0$
$a(z-w)+b(x-y)=0$
How can I approach such a problem? Are additional assumptions on $a,b$ helpful in this case?