Does the equation have at most one solution in integers?

Question

The question is whether an equation $\dfrac{x+y}{xy}+ \dfrac{1}{x^2+y^2}=\dfrac{1}{2}$ has more than one integer solution? The one integer solution is 1 and -1, so how can I prove that it is the only one? I tried to factor the equation to prove that there will be no even division but it didn't work out. Also was idea to assume that there are integer solutions and try all possible solutions (x-even and y-odd, x-odd and y-even, both x,y-odd or even) by writing them down as 2n or 2n+1 and to show that there will the contradiction but it didn't work either. Any help or hint would help a lot, thank you!

Answer 1

inequalities; graph the curve over the reals. There are two connected components. Note that $x \neq 2, y \neq 2.$

Next, when $ x < -1,$ we find $1 < y < 2,$ so $y$ cannot be an integer.

When $ x \geq 7,$ we find $2 < y < 3,$ so $y$ cannot be an integer.

You may solve explicitly for $y$ for each remaining $x$ value, $-1,0,1,2,3,4,5,6$ Let's see, when $x > 2$ we see $y > 2.$

Note that there is a little extra loop near the origin, but that the only integer point is the origin, which is not a legal $(x,y)$ position. So, the thing is not actually a hyperbola, but the inequalities are quite similar to that.