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In this work, we consider a very simple gravitational theory that contains ascalar field with its kinetic and potential terms minimally coupled to gravity,while the scalar field is assumed to have a coulombic form. In the context ofthis theory, we study an analytic, asymptotically flat, and regular(ultra-compact) black-hole solutions with non-trivial scalar hair of secondarytype. At first, we examine the properties of the static and sphericallysymmetric black-hole solution -- firstly appeared in 1504.08209 [gr-qc] -- andwe find that in the causal region of the spacetime the stress-energy tensor,needed to support our solution, satisfies the strong energy conditions. Then,by using the slow-rotating approximation, we generalize the static solutioninto a slowly rotating one, and we determine explicitly its angular velocity$\omega(r)$. We also find that the angular velocity of our ultra-compactsolution is always larger compared to the angular velocity of the correspondingequally massive slow-rotating Schwarzschild black hole. In addition, weinvestigate the axial perturbations of the derived solutions by determining theSchr\"{o}dinger-like equation and the effective potential. We show that thereis a region in the parameter space of the free parameters of our theory, whichallows for the existence of stable ultra-compact black hole solutions.Specifically, we calculate that the most compact and stable black hole solutionis 0.551 times smaller than the Schwarzschild one, while it rotates 2.491 timesfaster compared to the slow-rotating Schwarzschild black hole. Finally, wepresent without going into details the generalization of the derivedasymptotically flat solutions to asymptotically (A)dS solutions.

Analytic and asymptotically flat hairy (ultra-compact) black-hole solutions and their axial perturbations