| Abstract: |
| Regarding $N=1$ supersymmetric integrable equations, various supersymmetric generalizations of the Harry Dym equation, as well as the supersymmetric Schwarzian Korteweg-de Vries equation, fall into the $0$-homogeneous category. To comprehensively understand such equations, we classify $0$-homogeneous equations of the form
\begin{equation*}
U_t = U_{nx}F(U) + G\Big(U,(+\mathcal{D} U),U_{x},\cdots,U_{(n-1)x},(\mathcal{D} U_{(n-1)x})\Big),
\end{equation*}
where the super field $U=U(x,\theta,t)$ is bosonic, $U_{kx}(k\in\mathbb{Z}_{+})$ stands for the $k$th order derivative with respect to $x$ ($U_{1x}$ is abbreviated to $U_x$ as usual), and the super derivative $\mathcal{D}$ is defined as $\mathcal{D} = \partial_{\theta} + \theta\partial_x$. Eight $N=1$ supersymmetric integrable equations of such type are figured out by the symmetry approach, and most of them are shown to be connected with known ones by introducing appropriate changes of variables. |
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