The \textsc{ABC}-equation revisited: missed integrable cases. Part I: classification and (quasi) integrable structures

I. S. Krasil'shchik

We consider the \textsc{ABC}-equation $A{u}_x{u}_{yt}+B{u}_y{u}_{xt}+C{u}_t{u}_{xy}=0$, $A$, $B$, $C\in \mathbb{R}$, which was proved to be integrable in the case $A+B+C=0$ (the Ver…