  We classify Chabauty limits of groups fixed by various (abstract) involutions
over $SL(2,F)$, where $F$ is a finite field-extension of $\mathbb{Q}_p$, with
$p\neq 2$. To do so, we first classify abstract involutions over $SL(2,F)$ with
$F$ a quadratic extension of $\mathbb{Q}_p$, and prove $p$-adic polar
decompositions with respect to various subgroups of $p$-adic $SL_2$. Then we
classify Chabauty limits of: $SL(2, F) \subset SL(2,E)$ where $E$ is a
quadratic extension of $F$, of $SL(2,\mathbb{R}) \subset SL(2,\mathbb{C})$, and
of $H_\theta \subset SL(2,F)$, where $H_\theta$ is the fixed point group of an
$F$-involution $\theta$ over $SL(2,F)$.
