  In his beautiful paper [1], Ben Andrews obtained the complete classification
of the solutions of the planar isotropic $L_p$ Minkowski problem. In this
paper, by generalizing Ben Andrews's result we obtain the complete
classification of the solutions of the planar isotropic $L_p$ dual Minkowski
problem, that is, for any $p,q\in\mathbb{R}$ we obtain the complete
classification of the solutions of the following equation: \begin{equation*}
  u^{1-p}(u_{\theta}^2+u^2)^{\frac{q-2}{2}}(u_{\theta\theta}+u)=1\quad\text{on}\
\mathbb{S}^1. \end{equation*} To establish the classification, we convert the
ODE for the solution into an integral and study its asymptotic behavior,
duality and monotonicity.
