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 $G\mathrm{/}H$ 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 ${\mathbb{R}}^2$.