and which appear to be hidden in a purely numerical analysis.

When we speak about SFNSs and SFBHs, we are faced with some unsolved problems, such as the cosmic censorship conjecture, instability of configurations, violation of the nonnegativity of self-interaction potentials, and so on. However, a number of these problems, including the singularity problem, disappear if we assume that there is some additional matter (perhaps of an unknown nature, beyond the Standard Model) in the very centre. Note also that we regard, in what follows, a scalar field as a phenomenological construction rather than a fundamental natural field; on the other hand, the Standard Model of particle physics predicts the existence of such a field, and it is of great importance for the modern cosmology. In both the cases, we do not know (in advance, without appealing to astrophysical observations) the self-interaction potential of the scalar field. Therefore, in order to consider the problem in a sufficiently general approach, we use the so-called inverse problem method for self-gravitating spherically symmetric scalar fields; the method was proposed in [@Lechtenfeld1995; @Lechtenfeld1998; @BronnikovShikin2002] and later was explored in [@Tchemarina2009; @Azreg2010; @Cadoni2011; @Solovyev2012; @Cadoni2018] and applied, for example, in [@BronnikovChernakova2007; @Nikonov2008; @Franzin2018].

The paper 