SL(n + 1)-invariant equations which reduce to equations of Korteweg-de Vries type

Abstract

It is shown how to derive SL(n + 1)-invariant equations which reduce to scalar Lax equations for an operator of order n + 1. The existence of these systems explains the Miura transformation between modified Lax and scalar Lax equations. In particular we study an SL(2)-invariant system with a certain space of solutions lying over the solution space of a Korteweg-de Vries equation described by G. B. Segal and G. Wilson. This enables us to write down some solutions of this SL(2)-invariant system in terms of θ-functions of a hyperelliptic Riemann surface.