What are all real solutions of equation $x^x = y^y$? What about integer solutions and rational solutions? What are all real solutions of equation $x^x = y^y$? What about integer solutions and rational solutions?

For the case when $x=y$ it is clear, what about the case when $x\ne y$?
$x^x=y^y$ is obviously equivalent to $(1/y)^{1/x}=(1/x)^{1/y}$, and that's a very popular problem at MSE (e.g. $x^y = y^x$ for integers $x$ and $y$ containing also the general parametric solution).
&quot;OP&quot; asked the query:
$x^y=y^x$
Above has numerical integer solution:
$(x,y)=(2,4)$