Three-family conjecture for admissible transformations of two-dimensional shallow water equations

Let $\mathcal{C}$ denote the class of two-dimensional shallow water equations, let $G^\sim$ be its equivalence group, and call an admissible transformation independent when it is independent up to inversion and composition with other admissible transformations and with transformations generated by Lie symmetries of systems in $\mathcal{C}$. Three-family conjecture. The $G^\sim$-inequivalent non-identity admissible transformations of $\mathcal{C}$ that are independent in this sense are exhausted by the three families found in the paper. The conjecture concerns the explicit structure of the equivalence groupoid of the class, whose complete determination is described as difficult; the surrounding discussion also notes that several structural properties of admissible transformations remain to be proved.