In this talk, we present our recent work on the dynamics of a Kleinian group $\Gamma$ acting on the tangent bundle $T\mathbb{CP}^1$. Similar to the dynamics of $\Gamma$ on $\mathbb{CP}^1$, the action of $\Gamma$ on $T\mathbb{CP}^1$ partitions the space into a discontinuous part and a "recurrent" part lying above the limit set of $\Gamma$ in $\mathbb{CP}^1$. We explore the dynamics of $\Gamma$ on this recurrent part, with a focus on the case where $\Gamma$ contains a parabolic element. Additionally, we discuss the motivations behind these investigations and their applications, particularly in describing the dynamics of generalized semicomplete Halphen vector fields.
