Self-dual compact gauged baby skyrmions in a continuous medium

We investigate the existence of self-dual configurations in the restricted gauged baby Skyrme model enlarged with a $Z_2$--symmetry, which introduces a real scalar field. For such a purpose, we implement the Bogomol'nyi procedure that provides a lower bound for the energy and the respective self-dual equations whose solutions saturate such a bound. Aiming to solve the self-dual equations, we specifically focused on a class of topological structures called compacton. We obtain the corresponding numerical solutions within two distinct scenarios, each defined by a scalar field, allowing us to describe different magnetic media. Finally, we analyze how the compacton profiles change when immersed in each medium.