Classification of connected proper pairs in the affine transformation group

Let $(L, H)$ be closed subgroups of a locally compact group $G$. The pair $(L, H)$ is said to be proper if the action of $L$ on the homogeneous space $G/H$ is proper. We give a complete list of connected closed proper pairs in the affine transformation group of $\mathbb{R}^2$. This result extends Kobayashi's classification (1992) of connected closed subgroups of the affine transformation group of $\mathbb{R}^2$acting properly on $\mathbb{R}^2$.