The mumford dynamical system and the Gelfand–Dikii recursion

In his paper “The Mumford dynamical system and hyperelliptic Kleinian functions” [Funkts. Anal. Prilozhen. 57 (4), 27–45 (2023)] Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the $(P,Q)$-recursion introduced in his paper.
In the present paper, we further develop the theory of the $(P,Q)$-recursion and describe its connections to the Korteweg–de Vries hierarchy, the Lenard operator, and the Gelfand–Dikii recursion.