Abstract
We consider static, spherically symmetric, real scalar fields minimally coupled to Einstein gravity in the presence of electric charge. We present integral formulas exactly solving the inverse problem for a self-gravitating scalar field defined as a strictly monotonic function of a radial variable, so that one can find the associated field potential and spacetime metric using only integration. These formulas allow us to study some general properties of such scalar field configurations and also to find the two new classes of solutions: null-naked singularities and pointlike black holes. We discuss a classification of the possible solutions and give exact examples for all types of the asymptotically flat configurations. It is shown that there exist essential differences between the scalar and vacuum black holes in the behaviours of the orbital angular frequency, the radial velocity of a test particle falling towards the centre and in the locations of their innermost stable circular orbits.
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