Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$ 
Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

Obviously $(2,1)$ and $(1,2)$ are two answers. But I was unable to manipulate the equation algebraically giving a useful form for finding all other possible answers!
I also tried to view it as a quadratic equation in terms of $x$, but forming the Delta of that equation didn't help much other than if $y=1$ then certainly Delta would be a square and equation would have answer for $x$...

Perhaps it would be easier if you write the equation like this 
$$ xy(x+y)-(x+y)^2+2xy = 1$$ and now put $a=x+y$ and $b=xy$. Then you get: $$ ba-a^2+2b=1\implies b ={a^2+1\over a+2}$$
Write $c=a+2$ and then $$b = {(c-2)^2+1\over c} = {c^2-4c+5\over c}= c-4 +{5\over c}$$
so $c\mid 5\implies a+2\in \{-5,-1,1,5\} \implies a\in \{-7,-3,-1,3\}$ and respectively $b\in \{-10,-10,2,2\}$. This should not be difficult to finsih.
