127. Plot the graph of the equation

$$
\frac{1}{x^{2}+1}+\frac{1}{y^{2}+1}=\frac{2}{x y+1}
$$
Solution. By moving all terms of the equation to one side and combining like terms, we obtain the equation

$$
x^{2}+y^{2}-2 x y+2 x^{2} y^{2}-x^{3} y-x y^{3}=0
$$

or

$$
(x-y)^{2}-x y\left(x^{2}-2 x y+y^{2}\right)=0, (x-y)^{2}(1-x y)=0
$$

From this, $x=y$ or $x y=1$.

Therefore, the graph of the given equation is the union of the line $y=x$ and the hyperbola $y=\frac{1}{x}$.