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

We consider the \textsc{ABC}-equation $Au_xu_{yt} + Bu_yu_{xt} + Cu_tu_{xy} = 0$, $A$, $B$, $C\in\mathbb{R}$, which was proved to be integrable in the case $A + B + C = 0$ (the Veronese web equation). Our study reveals ten more cases that admit integrable structures that, to the best of our knowledg