Let $u(x,y)=f(x)+y$ be a quasilinear utility. Now we rotate it by 45 degrees, (such that the $x-$axis becomes the direction of $(1,1)$)
$v(x,y)=f(x-y)+x+y$. Is $v$ also a quasilinear utility? What is its name? It share many similarity with quasilinear utility such as parallel indifference curves. Being a rotation of quasilinear utility seems like the iff condition of parallel indifference curve.