TITLE: How prove this diophantine equation $(x^2-y)(y^2-x)=(x+y)^2$ have only three integer solution?
QUESTION [8 upvotes]: HAPPY NEW YEAR To Everyone! (Now Beijing time  00:00 (2015))

Let $x,y$ are integer numbers,and such $xy\neq 0$,
Find this diophantine equation all solution
$$(x^2-y)(y^2-x)=(x+y)^2$$

I use Wolf found this equation only have two  nonzero integer solution $(x,y)=(-1,1),(-1,-1)$,see wolf
$$\Longleftrightarrow x^3+y^3+x^2+y^2=xy(xy-1)$$
But How prove it?
and I found sometimes,and my problem almost similar with 2012 IMO shortlist:2012 IMO shortlist

REPLY [4 votes]: Please don't vote. Just a pictural comment to show what
the curve $(x^2-y)(y^2-x)-(x+y)^2=0$ does look like ( : it's an "elbow" ) .

$\color{red}{Red}$ is positive, $\color{green}{green}$ is negative.
The integer coordinates are yellow lines.
Picture on the right is $\color{blue}{blue}$ rectangle in picture on the left zoomed in $20 \times$ .