Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane

In \cite{LX}, the first author and Xu introduced and studied the horospherical $p$-Minkowski problem in hyperbolic space ${\mathbb{H}}^{n+1}$. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and $p\ge -n$. This paper focuses on the isotropic horospherical $p$-Minkowski problem in hyperbolic plane ${\mathbb{H}}^2$, which corresponds to the equation \begin{equation}\label{0} \varphi^{-p}\left(\varphi_{θθ}-\frac{\varphi_θ^2}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=γ\quad$\text{on}$\ $\mathbb{S}$^1, \end{equation} where $\gamma$ is a positive constant. We provide a classification of solutions to the above equation for $p\ge -7$, as well as a nonuniqueness result of solutions for $p<-7$. Furthermore, we extend this problem to the isotropic horospherical $q$-weighted $p$-Minkowski problem in hyperbolic plane and derive some uniqueness and nonuniqueness results.