VPV Identities related to ${\mathbf{x}}^{\mathbf{y}}={\mathbf{y}}^{\mathbf{x}}$ and ${\mathbf{x}}^{\mathbf{y}}{\mathbf{y}}^{\mathbf{x}}={\mathbf{v}}^{\mathbf{w}}{\mathbf{w}}^{\mathbf{v}}$

We cover rational and integer solutions for the equations ${\mathbf{x}}^{\mathbf{y}}={\mathbf{y}}^{\mathbf{x}}$ and ${\mathbf{x}}^{\mathbf{y}}{\mathbf{y}}^{\mathbf{x}}={\mathbf{v}}^{\mathbf{w}}{\mathbf{w}}^{\mathbf{v}}$. 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.