The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

Abstract Additional symmetries of the extended bigraded Toda hierarchy were introduced by Bakalov and Wheeless. We describe properties of a Lax operator fixed under these additional symmetries and determine the unique such Lax operator in the special case of the extended Toda hierarchy (ETH). We further present differential equations for the wave functions and their general solutions that hint at a possible connection to the bispectral problem. Based on these solutions, we highlight a correspondence between the wave functions. Finally, we find the form of a tau function for our Lax operator and compute a second solution to the ETH by applying a Darboux transformation presented by Li and Song.