Volterra-Bogoyavlensky lattices and solutions of $A_{2n}^{(1)}$ invariant Painlevé equations

The objective of this work is to develop a framework that exploits the lattice structure of the $k$-th Volterra--Bogoyavlensky equations ($k\in\mathbb N$, $k>1$) to generate rational solutions of higher symmetric Painlevé equations. For $k=2$, we show that the Volterra lattice, equipped with suita