x^2 + y^2 = 2(x+y) + xy

x      y

0     0

2     0

0     2

2     4

4     2

4     4

This may be a little difficult to solve algebraically  [ although there may be a way to do so ]

Look at the graph here :  https://www.desmos.com/calculator/oznsn0uwpe

Because of the presence of the xy term,  we have a rotated conic (ellipse??)  

We are looking for all the lattice points  (points with integer coordinates ) that this graph passes through

They are the  same ones I enumerated in my previous answer.......