Prove the equation has unique class of solutions Find the solutions of equation:
$$
x^y + y^x = 1 + xy \quad x,y \in \mathbb{R} \quad x,y &gt;0
$$

My quest

First, $x=1$ or $y=1$ gives us obvious solutions, so let's suppose $x \not =1$ and $y \not= 1$.

Using Bernoulli's inequality I was able to show the inequality:
$$
x^y + y^x &gt; 1 + xy \quad \forall x,y&gt;0 \quad and \quad (x-1)(y-1) \gt 0
$$
Therefore there is no solution with $x,y\gt1$ or $x,y\lt1$.

My goal is to show that there is no solution having $x \not =1$ and $y \not= 1$ and the only case I was not able to cover is $(x-1)(y-1)&lt;0$.

Any help will be appreciated.
you have given no source and no indication of your background. The conclusion that the solution set is the lines (in the first quadrant only) $y=1$ and $x=1$ seems quite likely given these plots; if you know multivariable calculus this gives some directions to pursue.
By the fundamental theorem on solutions to non linear equations, there are exactly n - 1 solutions, where n is the number of variables