Difficulties with SDiff(2) Toda equation (Q2781688)

This paper is devoted to the SDiff(2) Toda equation, which can be written for one unknown function of three variables, with differentiation shown by a subscript: NEWLINE\[NEWLINEu_{,z\overline{z}}= e^{u_{,tt}} \iff v_{,z\overline{z}}= (e^v)_{,tt}, \qquad v\equiv u_{,tt}. \tag{1}NEWLINE\]NEWLINE It is commonly accepted that to find a more direct method to ``buy'' new solutions from old ones begins with the Bäcklund transformation for the (2-dimensional) Toda lattice built over the Cartan matrix for the Lie algebra \(A_n\) or \(A_n^{(1)}\), and consider the limit as \(n\to\infty\). While this limit gives the derived PDEs, the associated limit of the (first-order) PDEs that define the Bäcklund transformations gives only trivial results. The author analyzes this phenomena as well as well-known ``Lax pairs''.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].