Junction conditions for local rotationally symmetric spacetimes in the $1+1+2$ covariant formalism

We use the distribution formalism to derive the complete set of junction conditions for general local rotationally symmetric (LRS) spacetimes in the $1+1+2$ covariant formalism. We start by developing a parametric framework encompassing timelike, spacelike, or null hypersurfaces. We then introduce the distribution formalism in the $1+1+2$ framework and obtain the necessary conditions to preserve the regularity of the $1+1+2$ equations at the separation hypersurface. Using these results, we can deduce some general prescriptions on the junction of LRS spacetimes and the properties of the shell in the nonsmooth cases. As examples of the application of the junction conditions, we use this formalism to perform the matching necessary to obtain well-known solutions, e.g., the Martinez thin shell, the Schwarzschild constant-density fluid star, and the Oppenheimer-Snyder collapse.