Field Sources for $f(R,R_{\mu\nu})$ Black-Bounce Solutions: The Case of K-Gravity

In the framework of Simpson-Visser, the search for field sources that produce black-bounces in alternative gravity theories has remained unresolved. In this paper, the first in a series exploring sources for alternative theories of gravity, we identify such a source for the $2+1$ dimensional K-gravity black-bounce. The K-gravity black hole is notable for allowing asymptotically locally flat solutions in lower-dimensional spacetime, yet it possesses curvature singularities concealed within the event horizon. Using the Simpson-Visser regularization technique, we eliminate this singularity, constructing asymptotically locally flat black-bounce solutions in $2+1$ dimensions. We explore the causal structure of these solutions, identifying the conditions under which they describe regular black holes or wormholes. By calculating curvature invariants, we confirm the absence of singularities within the event horizon. Additionally, we demonstrate that, beyond non-linear electrodynamics, a non-linear scalar field is required to source the solution. Finally, we investigate the geodesic structure of this spacetime, analyzing the trajectories of both massive and massless particles. We also confirm the existence of circular orbits and assess their stability.