  We cover rational and integer solutions for the equations $\mathbf{x^y =
y^x}$ and $\mathbf{x^y y^x = v^w w^v}$. The former equation solutions go back
to Euler, and the latter equation solutions appear to be new. Another
definitely new related topic is application of VPV identities to give
transforms of infinite products from our solutions. The present paper is
essentially chapter 28 of the author's book appearing in June 2024.
